[Video] Composition of Functions

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In this set of videos, we will work to understand composition of functions, including evaluation of composition of functions through function notation and graphs, building compositions of functions recognition a function composites.

The following videos will cover topics about composition of functions:

Introduction and Question 1: Definition of composition of functions and evaluation

(Click the arrow to the right to see the Introduction video and first question.)

(Click the arrow to the right to see the question posed at the end of the video.)

For $f(x) = x^2 + 1$ and $g(x) =2x-4$ , what is $(g \circ f)(2)$ ?
$(g\circ f)(2) \text { is exactly }\answer [given]{6}$

(Click the arrow to the right to see an example.)

Example 1

Question 2: Evaluation of compositions of functions via graphs.

(Click the arrow to the right to see the second question.)

(Click the arrow to the right to see the question posed at the end of the video.)

Given the graphs of the functions $f(x)$ and $g(x)$ below, what is $({\color {red} f} \circ {\color {blue} g})(1)$ ?

$({\color {red} f} \circ {\color {blue} g})(1) \text { is exactly }\answer [given]{1}$

(Click the arrow to the right to see an example.)

Example 2

Question 3: Recognizing the pieces of composite functions.

(Click the arrow to the right to see the second question.)

(Click the arrow to the right to see the question posed at the end of the video.)

Consider the function $h(x) = (x + 1)^2$ . If $h(x) = (f\circ g)(x)$ where $g(x) = x+1$ and $f(x)$ is unknown, what is $f(x)$ ?
$\text {The function }f(x) \text { is exactly }\answer [given]{x^2}$

(Click the arrow to the right to see an example.)

Example 3

Question 4: Determining pieces of a composite function.

(Click the arrow to the right to see the second question.)

(Click the arrow to the right to see the question posed at the end of the video.)

Consider the function $h(x) = x^2 + 4x - 7$ . If $h(x) = (f\circ g)(x)$ where $g(x) = x+1$ and $f(x) = ax^2 + bx + c$ for some values $a,b,$ and $c$ , what is $a, b,$ and $c$ ?
$a = \answer [given]{1}$ $b = \answer [given]{2}$ $c = \answer [given]{-10}$

(Click the arrow to the right to see an example.)

Example 4