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Formula for composite function $f$ of $g$ of $x$ .

$f(g(x))$

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Formula for composite function $f$ of $g$ of $x$ .

$f(g(x))$

What is the identity function?

$f(x) = x$

Formula for vertical translation up by $k$ units.

$f(x) + k$

Formula for vertical translation down by $k$ units.

$f(x) - k$

Formula for horizontal dilation (stretch) by a factor of $k$ where $0 < k < 1$ .

$f(kx)$

Formula for horizontal dilation (shrink) by a factor of $k$ where $k > 1$ .

$f(kx)$

How to find $f(g(x))$ given $f(x) = x + 1$ and $g(x) = x^2$ ?

Replace $x$ in $f(x)$ with $g(x)$ : $f(g(x)) = (x^2) + 1 = x^2 + 1$ .

How to evaluate $f(g(2))$ given $f(x) = 2x - 1$ and $g(x) = x^2 + 3$ ?

First, find $g(2) = 2^2 + 3 = 7$ . Then, find $f(7) = 2(7) - 1 = 13$ .

How to decompose $h(x) = (x + 2)^2$ into two functions, $f(x)$ and $g(x)$ ?

Let $g(x) = x + 2$ and $f(x) = x^2$ . Then $f(g(x)) = (x + 2)^2 = h(x)$ .

How to find $x$ such that $f(g(x)) = 5$ , given $f(x) = x + 1$ and $g(x) = 2x$ ?

First find $f(g(x)) = 2x + 1$ . Then solve $2x + 1 = 5$ , which gives $x = 2$ .

How to find $f(g(x))$ if $f(x) = x^2$ and $g(x) = \sqrt{x}$ ?

Substitute $g(x)$ into $f(x)$ : $f(g(x)) = (\sqrt{x})^2 = x$ , for $x \geq 0$ .

How to determine the domain of $f(g(x))$ ?

Find the domain of $g(x)$ and ensure that the range of $g(x)$ is within the domain of $f(x)$ .

Explain the order of operations in $f(g(x))$ .

Evaluate $g(x)$ first, then use the result as the input for $f(x)$ . Work from the inside out.

Why is the order important in composite functions?

Composition is generally not commutative; $f(g(x))$ is usually not equal to $g(f(x))$ .

What happens when you compose a function with the identity function?

The result is the original function; $f(x) = x$ implies $g(f(x)) = f(g(x)) = g(x)$ .

How do you find $f(g(x))$ analytically?

Substitute the entire function $g(x)$ for every instance of $x$ in $f(x)$ .

How do composite functions relate to transformations?

They can represent transformations such as vertical translations ( $f(x) + k$ ) and horizontal dilations ( $f(kx)$ ).

Explain how to use graphs to evaluate composite functions.

Find the output of $g(x)$ from its graph, then use that output as the input for $f(x)$ on its graph.