Notes
Functions, domain and range
- A function gives exactly one output for each input. Find the domain by excluding values that make a denominator zero or put a negative under an even root.
- The range is the set of outputs. For example, $x^2 + 4$ has range $[4, \infty)$.
Transformations and symmetry
- $f(x - h) + k$ shifts right $h$ and up $k$. $-f(x)$ flips over the x-axis, and $f(-x)$ flips over the y-axis.
- Even functions satisfy $f(-x) = f(x)$ (y-axis symmetry). Odd functions satisfy $f(-x) = -f(x)$ (origin symmetry).
Combining, composing and inverting
- $(f \circ g)(x) = f(g(x))$: apply $g$ first. Order matters.
- To find $f^{-1}$, swap $x$ and $y$ and solve for $y$. Only one-to-one functions have inverses, and $f(a) = b$ means $f^{-1}(b) = a$.
- Average rate of change from $a$ to $b$ is $\displaystyle \frac{f(b) - f(a)}{b - a}$.
Topics in this unit
Each topic has its own short notes and practice.
Practice
The root needs $4 - x \ge 0$ and it is in the denominator, so $4 - x > 0$.
The graph shifts right 3 and up 1; the stretch by 2 does not move the vertex.
$f(-x) = -x^3 + 4x = -f(x)$.
$g(5) = 2$, then $f(2) = 4 + 1 = 5$.
Swap: $x = y^3 + 2$, so $\displaystyle y = \sqrt[3]{x - 2}$.
$\displaystyle \frac{f(3) - f(1)}{3 - 1} = \frac{27 - 1}{2} = 13$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Let $f(x) = \sqrt{x + 2} - 1$.
(a) State the domain and range of $f$.
Show the answer and scoring guide
The radicand must be nonnegative: $x + 2 \ge 0$, so the domain is $[-2, \infty)$. Since $\displaystyle \sqrt{x+2} \ge 0$, the range is $[-1, \infty)$.
Check off the points you earned:
(b) Find $f^{-1}(x)$ and state its domain.
Show the answer and scoring guide
Write $\displaystyle y = \sqrt{x + 2} - 1$ and swap: $\displaystyle x = \sqrt{y + 2} - 1$. Then $(x + 1)^2 = y + 2$, so $f^{-1}(x) = (x + 1)^2 - 2$ with domain $x \ge -1$ (the range of $f$).
Check off the points you earned:
(c) Verify your inverse using $x = 7$.
Show the answer and scoring guide
$\displaystyle f(7) = \sqrt{9} - 1 = 2$. Then $f^{-1}(2) = (2 + 1)^2 - 2 = 7$, which returns the original input.
Check off the points you earned: