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What is a composite function?
A function formed by applying one function to the results of another: $f(g(x))$.
What does $f(g(x))$ mean?
Apply $g$ to $x$ first, then apply $f$ to the result.
What is the identity function?
The function $f(x) = x$, which returns the input unchanged.
Define function decomposition.
Breaking down a complex function into simpler component functions.
What is vertical translation in terms of function composition?
Shifting the graph of a function up or down by adding a constant, represented by $f(x) + k$.
What is horizontal dilation in terms of function composition?
Stretching or shrinking the graph of a function horizontally by multiplying the input by a constant, represented by $f(kx)$.
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 $f(x)$.