[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.)

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

Question 2: Evaluation of compositions of functions via graphs.

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

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

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

Question 3: Recognizing the pieces of composite functions.

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

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 $f(x)$ ?
$\text {The function }f(x) \text { is exactly }\answer [given]{x^2}$

Question 4: Determining pieces of a composite function.

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

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 $c$ ?
$a = \answer [given]{1}$ $b = \answer [given]{2}$ $c = \answer [given]{-10}$

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sym_name	$\backslash\texttt{\blue{[?]}}$
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int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
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omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

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