Khan Academy · YouTube · 6:14
This introductory algebra whiteboard clip explains function composition by first showing three ways to define functions: a formula for , a table for , and a graph for . The speaker then works the example , stressing that one must evaluate the inner function first. From the table, ; substituting into f gives (-3)^. A simple flowchart visually encodes the inside-out evaluation order. This video segment demonstrates how to evaluate composite functions, specifically and . The function f is defined algebraically as . The function g is defined by a table of values, and h is defined by a graph. The presenter shows two methods for evaluating : a direct algebraic substitution and a visual flow diagram. Both methods correctly yield the result 0. The segment concludes with the presenter proposing a more complex composition, , but the solution is not shown. This 74-second whiteboard segment works one concrete example of function composition: evaluating . The board presents three different representations of functions at once: f by the formula , g by a table, and h by a graph. The speaker explains that this is a “triple composition,” then evaluates from the inside out: first , then from the table, and finally from the graph. The result written and spoken is . The clip’s main teaching point is procedural: nested compositions are simplified step by step by feeding each output into the next function.
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Generated from the video's visuals and explanation; not verbatim speech.
The board opens with three different kinds of function definitions placed side by side. On the left is the formula . In the middle is a table with columns and , listing , , , and . On the right is a graph labeled . The speaker uses these to show that a function can be given algebraically, by a table, or by a graph.
While pointing to the graph, the speaker extracts two sample values: and . This reinforces that a graph also defines an input-output rule: choose an -value and read the corresponding -value.
The lesson then shifts to composition. The speaker explains that composing functions means building one function out of others, or nesting them. The key idea is not a new operation on numbers, but a new order of evaluation: the output of one function becomes the input of another.
As a warm-up, the expression is written on the board. The speaker acknowledges that the notation may look intimidating at first, then reduces it to the basic definition of a function as a mapping from one number to another.
A flowchart is drawn to make the process concrete: . This visual sequence shows the inside-out rule for composition. First the input 2 goes into ; the result then goes into .
The inner value is found from the table. Since the row with has , the speaker concludes . This is the crucial intermediate result needed before the outer function can be evaluated.
Now the outer function is applied. Substituting into gives . The arithmetic is simplified on the board to , so the final value of the composite expression is .
Let's review how we evaluated the composite function . We first found that equals -3 from the table. Then, we plugged this value into f, calculating as (-3)^2 - 1, which simplifies to 9 - 1, giving a final answer of 8.
Now, let's apply the same logic to a new problem: what is the value of ? I encourage you to pause and try it yourself before we proceed.
We can solve this algebraically. The function f is defined as . In the expression , the entire term acts as the input 'x'. So, we substitute into the formula, getting ()^2 - 1.
To finish the calculation, we need the value of . Looking at the graph of , we find the point where . The corresponding y-value is 1, so .
Substituting this back into our expression, we get (1)^2 - 1, which is 1 - 1, resulting in 0. Therefore, .
Alternatively, we can visualize this process with a flow diagram. We start with our initial input, 2. An arrow points to a box for the inner function, h. The output of is 1, so we write 1 next.
From there, another arrow points to a box for the outer function, f. We now evaluate . Using the formula for f, we calculate , which equals 0. This is our final output.
Both the algebraic method and the flow diagram confirm that is 0. The diagram clearly shows that the output of the first function becomes the input for the second.
We can extend this concept further by composing three functions. Let's consider the expression . This means we will evaluate the functions from the inside out: first , then use that result in another f, and finally use that second result as the input for g.
Let's pause here and think about how we would begin to solve . What would be the very first step?
The clip opens on a blackboard already containing three function representations: the formula , a table for , and a graph labeled . Earlier composite examples remain visible, so the viewer sees that the lesson is about chaining functions rather than studying one function in isolation.
The instructor introduces a new expression, , and calls it a triple composition. Mathematically, this means the output of at 2 becomes the input of , and the output of that step becomes the input of .
The chosen method is to start from the innermost layer. Using the displayed rule for , the instructor computes . This substitution reduces the original expression to .
Next, the middle function is handled by table lookup rather than formula manipulation. Since the table shows the row paired with , the instructor obtains . The composition now simplifies further to .
For the outermost function, the instructor returns to the graph of . Reading vertically from to the plotted curve and then horizontally to the -axis gives . Therefore the full nested expression evaluates to .
The closing remark frames the example as a general procedure: to evaluate a composition, resolve the inside function first, pass its result to the next function, and continue outward until a single numerical output remains.
The clip begins by showing that a function can be defined in multiple ways: by an algebraic formula, by a table of input-output pairs, or by a graph. Here is given by a rule, by a table, and by a picture of its graph.
When a function is given as a graph, an input value is located on the horizontal axis and the corresponding output is read from the curve. The speaker demonstrates this with the graph of , noting and .
Composition means using the output of one function as the input of another. Informally, the speaker describes it as nesting functions, so that evaluating a composite expression proceeds from the inside outward.
To demystify composite notation, the speaker reminds the viewer that a function is simply a mapping from one number to another. Thus is not mysterious: it is just two ordinary mappings performed in sequence.
The flowchart shows the exact order of operations. First compute the inner value , then feed that result into the outer function .
Because is defined by a table, the value is found by locating the row where . The matching output is , so .
After obtaining the inner value, substitute it into the formula for . With , the outer evaluation becomes , which simplifies to .
Putting the steps together, the example shows that the composite expression evaluates to 8. This is the central worked result of the clip.
To evaluate a composite function like , you work from the inside out. First, find the value of the inner function, . Then, use that result as the new input for the outer function, f. This method applies regardless of whether the functions are defined by formulas, tables, or graphs.
When a function is defined by a table, like , you find the function's value by looking up the input in the first column and reading the corresponding output from the second column. For example, to find , you locate '2' in the 't' column and see that the value in the '' column is '-3'.
When a function is defined by a graph, like , you find the function's value by locating the input on the horizontal axis (x-axis), moving vertically to the graph, and then moving horizontally to the vertical axis (y-axis) to read the output. For example, to find , you locate , move up to the graph, and see that the corresponding y-value is 1, so .
A flow diagram is a visual tool for evaluating composite functions. You start with the initial input value, draw an arrow to a box representing the first (innermost) function, write the output of that function, draw another arrow to a box for the next function, and so on. This clearly illustrates the sequence of transformations from the original input to the final output.
The expression is a composition of three functions. Evaluation proceeds from the inside outward: first compute , then use that result as the input to , and finally use the result of as the input to .
When a composite expression contains several nested functions, the demonstrated method is to evaluate the innermost one first. In this clip, that means computing before dealing with or .
If a function is defined by a table, find the row whose input matches the desired argument and read the corresponding output. Here the table for gives because the row with has output 4.
If a function is defined by a graph, locate the input on the horizontal axis and read the -coordinate of the graph at that -value. In the clip, the graph of shows that .
By chaining the substitutions , , and , the original triple composition reduces to a single number: .
A learner may mistakenly try to begin with the leftmost function symbol in a nested expression. The clip shows the opposite rule: for , the correct order is innermost first, then middle, then outermost.
Explore conditions, steps and evidence. Supplementary explanations are labeled separately from content shown in the video.
The board shows in the upper left.
The speaker refers to this as one of the three different function definitions.
A function defined by the rule that the input is squared and then 1 is subtracted.
The video does not explicitly state a domain; the example evaluates f at -3.
A table with columns t and is shown on the board.
The speaker says the table maps between values of t and what would be.
The output value of the function g when the input is t, given by the displayed table.
The displayed table explicitly lists inputs , 2, 3, 4.
A graph labeled is shown on the right side of the board.
The speaker says this maps from x to .
The output value of the function h at input x, read from the graph.
The visible graph spans approximately to .
The expression is written on the board.
The speaker introduces it as a warm-up example for composing functions.
First evaluate g at 2, then use that result as the input to f.
Defined here because is listed in the table and f accepts that output.
x appears in and .
The speaker uses x as the input variable for f and h.
x
Input variable for the functions f and h in this clip.
No explicit domain is stated beyond the examples shown.
t appears as the first column header in the table for .
The speaker says the table maps between different values of t and .
t
Input variable for the function g in the displayed table.
Explicitly shown values are 1, 2, 3, 4.
The board shows .
A function defined by squaring the input and subtracting 1.
All real numbers (implied by the polynomial expression).
A table is shown with columns t and , listing pairs (1, 3), (2, -3), (3, 4), (4, -1).
A function defined by a discrete table of values.
The set {1, 2, 3, 4} based on the visible table.
A graph labeled is shown. The speaker reads from it.
A function defined by its graph.
The visible x-range of the graph, approximately [-4, 4].
Used as the independent variable in and as the horizontal axis label for .
x
The independent variable or input to a function.
Used as the header for the first column of the table defining .
t
The independent variable or input for the function g in the table.
The board shows .
The speaker evaluates by substituting into this formula.
f
A function defined on the displayed board by the rule .
The video does not explicitly state a domain; in the worked example it is applied at .
The speaker says there are three different function definitions and describes each representation.
The board simultaneously shows , a table for , and a graph for .
The clip presents three equivalent ways to specify a function: an algebraic rule for f, a finite input-output table for g, and a graph for h. The speaker treats all three as mappings from inputs to outputs.
Each representation must assign outputs to inputs.
The speaker says, 'A function is just a mapping from one set of numbers to another.'
Before evaluating the composite expression, the speaker restates the basic idea that a function takes an input number and produces an output number according to its rule.
There must be an input value and a rule or representation that determines the output.
The speaker introduces composing functions as building up a function by composing one function of other functions, or nesting them.
Composition means using the output of one function as the input of another. The speaker frames this as nesting functions inside one another.
The inner function must produce a value that can be used as an input to the outer function.
The speaker explains that gives an output, and then that output is input into f to get .
A flow diagram is drawn: 2 -> [g] -> -> [f] -> .
To evaluate , first compute the inner value . Then substitute that result into the outer function f. The visual flowchart makes the order of operations explicit.
must be defined.
The value must lie in the domain of f for the composition to be evaluated.
The table shows t values 1, 2, 3, 4 with corresponding values 3, -3, 4, -1.
The speaker says when t is equal to 2, is negative 3.
For a function defined by a table, locate the input row and read the matching output. Here corresponds to , so .
The desired input must appear in the table.
The speaker substitutes -3 into f and computes (-3)^.
The board shows .
Once the inner value is known, replace x in the formula for f with that value. Since , the outer evaluation becomes .
The substituted value must be a valid input for f.
The speaker gives examples from the graph: when , ; when , .
The graph labeled is visible on the right side of the board.
The exact full piecewise formula for is not written out in this clip.
For a graph-defined function, find the input on the horizontal axis and read the corresponding output from the plotted curve. The speaker demonstrates this with and .
The graph must define a single output for each input considered.
The speaker explains how to evaluate and by first evaluating the inner function and then using that result as the input to the outer function.
The board shows the step-by-step evaluation: , and .
To evaluate a composite function like , you first find the value of the inner function, . Then, you use that resulting value as the new input for the outer function, f. This process can be extended to compositions of three or more functions, working from the innermost function outwards.
The output of the inner function must be within the domain of the outer function.
The speaker draws a flow diagram: 2 -> [h] -> 1 -> [f] -> 0 to represent the evaluation of .
The speaker says, 'we could have done it with the diagram way... we're going to input 2 into h... you get 1... and then we're going to input that into f...'
A visual method for understanding function composition. You start with the initial input value, draw an arrow to a box representing the first (innermost) function, write the output of that function, draw another arrow to a box for the next function, and so on until you reach the final output. This clearly shows the sequence of operations.
The speaker introduces and says, 'So now we're really doing a triple composition.'
The board displays the nested expression .
The clip defines the task as evaluating a nested composition in which one function output becomes the next function's input. Here three functions are chained: f is evaluated first, its result is fed into g, and that result is fed into h.
Each intermediate output must lie in the domain of the next function.
Evaluation proceeds from the innermost function outward.
The speaker says one way is to evaluate what is , then computes 2 squared minus 1 to get 3.
The board shows written above the inner part of .
For a nested expression like , the method demonstrated is to start with the innermost function , replace it by its numerical value, and then continue outward. This reduces the composite step by step instead of trying to simplify the whole expression at once.
The function rule or representation for the inner function must be available.
The speaker asks 'Now what is g of 3?' and answers that when t is equal to 3, is 4.
The table on the board lists t values 1, 2, 3, 4 with corresponding values 3, -3, 4, -1.
When a function is given by a table, the value is found by locating the row where the input column t equals 3 and reading the matching output in the column. The clip uses this to replace by 4 inside the composition.
The desired input must appear among the listed table inputs.
The speaker states that composing functions means building up a function by composing one function of other functions, or nesting them.
Composing functions means forming a new evaluation process by putting the output of one function into another function.
Two or more functions are available for evaluation.
Informal universal statement about the meaning of composition in this lesson.
The table entry for is .
The speaker says is negative 3.
From the displayed table, .
The table defines g at .
Specific instance.
The board shows .
The speaker concludes that this is equal to 8.
Using the given definitions of f and g, .
.
from the table.
Specific instance.
The speaker walks through the composition verbally from inner function to outer function.
A flowchart is drawn showing 2 entering g, producing , then entering f to produce .
The final written computation is .
Start with the composite expression introduced on the board.
This is the example the speaker chooses to demonstrate function composition.
Evaluate the inner function first by sending the input 2 into g.
By the definition of composition, the output of the inner function becomes the input of the outer function.
Read the value of g at input 2 from the table.
The table explicitly pairs with .
Substitute the computed inner value into the outer function.
Since , the outer function receives -3 as its input.
Apply the rule for f to the input -3.
The function f is defined by .
Square -3.
Arithmetic evaluation of the power.
Subtract 1.
Arithmetic simplification.
Therefore .
The board shows the calculation .
The speaker states, 'f of g of 2 is going to be equal to 8.'
Identify the inner function to evaluate first.
Order of operations for function composition.
Find the value of from the provided table.
Reading a value from a function defined by a table.
Substitute the result from the previous step as the input to the outer function f.
Definition of composite function .
Apply the rule for the function with .
Substitution into the function's formula.
Perform the arithmetic calculation.
Standard arithmetic.
The board shows the algebraic steps: .
The speaker explains, 'the input is h of 2... we're going to take the input which is h of 2 and we're going to square it and then we're going to subtract 1... Now what is h of 2? When x is equal to 2, h of 2 is 1. So since h of 2 is equal to 1, this simplifies to 1 squared minus 1, well that's just going to be 1 minus 1, which is equal to 0.'
Start with the composite function expression.
Problem statement.
Apply the definition of , where the input x is replaced by the entire expression .
Substitution into the function's formula.
Determine the value of the inner function by reading the point (2, 1) from the graph of .
Reading a function value from its graph.
Substitute the value into the expression from step 2.
Substitution.
Perform the final arithmetic calculation.
Standard arithmetic.
The speaker draws the flow diagram 2 -> [h] -> 1 -> [f] -> 0.
The speaker narrates the process: 'input 2 into h... you get 1... and then we're going to input that into f... which is going to give us f of 1. f of 1 is 1 squared minus 1, which is 0.'
Start with the initial input 2 and apply the innermost function h.
Flow diagram method for composition.
The output of is 1, found from the graph.
Reading a value from a graph.
Take the output 1 and use it as the input for the next function, f.
Flow diagram method for composition.
Calculate using the formula , which gives .
Substitution and arithmetic.
The final output of the composition is 0.
The speaker evaluates , then , then , and states the final result is -1.
The board successively annotates with 3 above , 4 below , and finally writes = -1.
The graph of h is referenced when determining .
Start with the full nested expression introduced by the speaker as a triple composition.
This is the problem statement written on the board.
Evaluate the innermost function first using the rule for f.
Substitution into the displayed formula .
Replace by 3 inside the composition.
Equal substitution of a computed value into the outer expression.
Read the next function value from the table for g.
The table entry with has .
Replace by 4, leaving only the outermost function to evaluate.
Equal substitution of the table value into the remaining composition.
Read the final value from the graph of h at .
The point on the graph above has y-coordinate -1.
Conclude the value of the original triple composition.
Chaining the previous substitutions yields the final numeric result.
The composite value equals -1.
The speaker calls this a warm-up and asks what will be.
The board shows , the table for , and the worked result .
Given and the table defining , evaluate the composite expression .
.
The table for includes the row , .
Compute the numerical value of .
Find the inner function value first from the table.
Composition requires evaluating the inner function before the outer one.
Replace by -3 inside f.
Substitution of the previously computed inner output.
Use the formula for f with input -3.
Direct application of .
Simplify the arithmetic to obtain the final value.
Standard arithmetic evaluation.
8
The result matches the final written line on the board: .
The speaker says when , , and when , .
A graph labeled is shown on the right.
Only selected points are discussed; the full graph-based rule for h is not fully transcribed in this clip.
Use the graph of to identify a few output values.
A graph labeled is displayed.
Extract example function values from the graph.
At , the graph gives output 0.
The speaker explicitly states this point from the graph.
At , the graph gives output 2.
The speaker explicitly states this point from the graph.
Sample values include and .
These values are directly spoken while the graph is on screen.
The problem is written on the board, along with its solution steps and final answer 8.
Evaluate the composite function .
Table for : , , ,
Find the numerical value of .
Look up the value of in the provided table.
Definition of g by table.
Substitute the value of into the outer function f.
Method for evaluating composite functions.
Apply the formula for with and calculate the result.
Substitution and arithmetic.
8
The calculation is shown step-by-step on the board and matches the speaker's narration.
The problem is written and solved on the board using two different methods.
The graph of is used to find , and a flow diagram is drawn to illustrate the second method.
Evaluate the composite function .
Graph of , from which can be read.
Find the numerical value of .
Locate on the graph of and find the corresponding y-value, which is 1.
Reading a function value from a graph.
Substitute the value of into the outer function f.
Method for evaluating composite functions.
Apply the formula for with and calculate the result.
Substitution and arithmetic.
0
The result is confirmed by two different methods shown on the board: direct algebraic substitution and a visual flow diagram.
The speaker proposes 'just for fun' and works it out completely.
The board shows the expression and the intermediate substitutions 3, 4, and final -1.
The graph of h is used at the last step.
Given , a table for g, and a graph for h, evaluate the triple composition .
The table for g includes .
The graph of h gives .
Find the numerical value of .
Compute the innermost value first.
Direct substitution into the formula for f.
Use the result 3 as the input to g and read the table.
Table lookup at .
Use the result 4 as the input to h and read the graph.
Graphical evaluation at .
Combine the successive replacements to obtain the final answer.
Composition is evaluated from inside out.
-1
The final board annotation matches the spoken conclusion that is negative 1.
The board is divided into three regions: formula for f at upper left, table for g near center, graph for h at right.
The speaker points out that these are three different function definitions.
Formula
Table with columns t and
Graph labeled
The cursor moves among the three representations as the speaker describes them.
All three objects remain visible as separate examples of function definitions.
The visual layout establishes that functions may be specified algebraically, by a table, or by a graph.
A chain is drawn: 2 -> [g] -> -> [f] -> .
The speaker narrates taking 2 into g, getting , then inputting that into f.
Input 2
Box labeled g
Intermediate output
Box labeled f
Final output
The diagram is built left to right, showing the order of evaluation.
The direction of arrows always points from input toward output.
The animation encodes the rule that composition is evaluated from the inside outward.
The board displays .
Composite expression
Substituted value -3
Intermediate arithmetic 9 - 1
Final answer 8
The expression is simplified step by step until the final numeric answer appears.
The outer function remains f and the inner value remains -3 throughout the simplification.
The visual sequence confirms the computational result of the composition example.
A green table with headers 't' and '' is displayed, showing four input-output pairs.
Table with columns t and
The values in the table remain constant throughout the clip.
This table provides a discrete definition for the function g. To find for a given t in the first column, one reads the corresponding value from the second column.
A yellow graph of a piecewise linear function labeled is displayed on a coordinate plane.
Coordinate plane
Graph of
The shape and key points of the graph remain constant.
This graph provides a continuous definition for the function h. To find for a given x, one locates x on the horizontal axis, moves vertically to the graph, and then horizontally to the vertical axis to read the corresponding y-value. For example, the point (2, 1) is on the graph, so .
The speaker draws a flow diagram step-by-step: '2', an arrow, a box with 'h', an arrow, '1', an arrow, a box with 'f', an arrow, and finally '0'.
Numbers 2, 1, 0
Boxes labeled 'h' and 'f'
Arrows connecting them
The diagram is constructed sequentially from left to right as the speaker explains the evaluation process.
This animation visually represents the order of operations in function composition. The initial input '2' is processed by function 'h' to produce '1'. This intermediate result '1' is then processed by function 'f' to produce the final output '0'.
The screen shows three representations side by side: the formula , a table for , and a graph labeled .
Below them are earlier composite examples including and , plus flow-style arrows showing input passing through boxes labeled g, f, and h.
Formula for f
Table for g
Graph for h
Earlier composite examples
Arrow-style evaluation diagrams
The new expression is written beneath the earlier examples.
Intermediate values 3 and 4 are added around the nested expression.
The final equality = -1 is appended.
The definitions of f, g, and h remain visible throughout the clip.
The earlier examples stay on the board as context while the new example is solved.
The visual arrangement supports the lesson by placing each function in a different representational form, so the learner sees that composition can combine formula-based, table-based, and graph-based functions.
As the speaker talks, the board adds 3 above , then 4 below , and finally writes = -1 for the whole expression.
The nested expression remains visible while annotations accumulate.
Expression
Number 3
Number 4
Final result -1
First the inner value is replaced conceptually by 3.
Then the middle value is replaced by 4.
Finally the outer value is replaced by -1.
The order of nesting does not change.
Each annotation corresponds to one function layer being resolved.
The animation visually encodes the inside-out evaluation procedure for nested functions.
The speaker says the notation seems daunting at first if you are not familiar with it, but one should remember what a function is.
Students may think is a new kind of object rather than ordinary sequential function evaluation.
The clip clarifies that still means: first apply g to 2, then apply f to the resulting number.
The speaker explicitly begins with 'what is f of 2' before moving to g and then h.
The board resolves the innermost term first and only then the outer terms.
A learner might try to start with the leftmost symbol h in and treat the expression as if the outer function should be handled first.
The clip demonstrates that composition is evaluated from the inside out: compute first, then use that result as the input to g, and only then apply h.
The speaker first recalls that a function is a mapping, then uses that idea to explain .
Understanding composition in this clip depends on treating each function as an input-output mapping.
The table for is used directly to obtain in the worked example.
The method of reading a table is applied to compute the inner value in the composite example.
The rule is used to evaluate in the example.
The algebraic substitution method is applied to finish the composite evaluation.
The board juxtaposes a formula, a table, and a graph as three definitions of functions.
Reading values from a graph is one specific instance of the broader idea that functions can be defined in multiple representations.
The speaker uses the table for g and the graph for h as inputs to the algebraically defined function f.
The method of evaluating composite functions is applied using functions defined in different ways (algebraic formula, table, and graph).
The speaker uses the table for g and the graph for h as inputs to the algebraically defined function f.
The method of evaluating composite functions is applied using functions defined in different ways (algebraic formula, table, and graph).
The speaker introduces the flow diagram as an alternative way ('the diagram way') to perform the same evaluation as the algebraic method.
The flow diagram method is a visual representation that is equivalent to the standard algebraic method for evaluating composite functions.
The example combines a formula for f, a table for g, and a graph for h in one computation.
The speaker switches among these representations while solving .
The general idea of function composition is applied through a concrete inside-out evaluation procedure.
The table for g is consulted after is computed.
The graph for h is consulted after is computed.
In this worked example, obtaining is needed before one can evaluate from the graph.
The board accumulates substitutions 3, then 4, then the final equality -1.
The final numeric conclusion depends on the earlier inside-out evaluation steps.
The speaker explicitly introduces the idea of composing functions.
The worked example on the board evaluates .
The table for is used to find .
The speaker reads and from the graph.
The entire clip demonstrates how to evaluate composite functions like and .
The value is read from the table.
The value is read from the graph.
A flow diagram is drawn to explain the evaluation of .
The speaker says, 'So now we're really doing a triple composition.'
The speaker begins by asking what is.
The speaker reads from the table as 4.
The speaker says to look back to the original graph and reads .
Covered · The clip introduces three function representations and gives sample readings from the graph of h.
Covered · The speaker defines composition informally as nesting one function inside another.
Covered · The example is posed, and the speaker reassures that the notation is just sequential mapping.
Covered · A flow diagram shows the order of evaluation for the composite expression.
Covered · The inner value is read from the table as -3.
Covered · The outer function is evaluated and the final answer 8 is obtained.
Covered · Evaluation of using the table for g.
Covered · Algebraic evaluation of using the graph for h.
Covered · Diagrammatic evaluation of .
Covered · Speaker proposes a new problem, , but does not solve it within the clip.
Covered · Opening audio continues from a prior example while the board already shows f, g, h and earlier compositions; the clip transitions into introducing .
Covered · The speaker explicitly names the new task as a triple composition and writes .
Covered · The innermost function is evaluated as and annotated on the board.
Covered · The value is read from the table as 4 and substituted into the composition.
Covered · The speaker reads from the graph and concludes .
Covered · Closing remarks summarize that the example demonstrates how to evaluate the composition of functions.
Reviewed English material v1: chapter "Introducing composition" at 0:30 — The lesson explains that composing functions means building one function out of others, i.e. nesting them, and that the key idea is a new order of evaluation in which the output of one function becomes the input of another, so a composite such as is evaluated from the inside outward.
YouTube10:15Next in Category Theory: Function composition → Categories.
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