Function Composition and Inverses: math practice, Grade 12

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Math practice Grade 12 : Function Composition and Inverses — Zyro the alien explorer of Planète Maths

24 problems graded by difficulty (★ easy, ★★ medium, ★★★ hard). Try first, without looking at the solutions!

Sommaire

1 Evaluating composites ★★★

Let $f(x)=2x+5$ and $g(x)=x^2-3$. Compute $(f\circ g)(3)$, $(g\circ f)(3)$ and $(f\circ f)(-1)$.

2 Building composite formulas ★★★

Let $f(x)=x-7$ and $g(x)=4x$. Find $(f\circ g)(x)$ and $(g\circ f)(x)$. Are they equal?

3 A simple domain ★★★

Let $f(x)=\sqrt{x}$ and $g(x)=x+6$. Find $(f\circ g)(x)$ and $(g\circ f)(x)$, and give the domain of each.

4 Horizontal line test ★★★

Decide which of these functions are one-to-one, and justify with the horizontal line test: $f(x)=2x-9$, $g(x)=x^2$, $h(x)=x^3+1$, $k(x)=|x|$, $m(x)=\dfrac1x$.

5 A linear inverse ★★★

Find the inverse of $f(x)=5x-8$, then check your answer with the input $7$.

6 Even, odd or neither ★★★

Classify each function as even, odd, or neither: $f(x)=x^6-2x^2$, $g(x)=4x^3+x$, $h(x)=x^2+3x$.

7 Absolute values and pieces ★★★

(a) Solve $|x+4|=9$.

(b) Let $q(x)=3x-1$ if $x\le2$, and $q(x)=x^2-1$ if $x\gt2$. Find $q(-2)$, $q(2)$ and $q(4)$.

8 Payroll at a repair shop ★★★

A phone repair shop schedules $h(d)=6d$ work hours over $d$ days. A technician earns $P(h)=22h+15$ dollars for $h$ hours (including a $15 bonus). Find $P(h(d))$ and use it to find the pay for $5$ days.

9 A hidden restriction ★★★

Let $f(x)=\dfrac{1}{x-2}$ and $g(x)=x+5$. Write $(f\circ g)(x)$ and find its domain.

10 Inverse of a rational function ★★★

Find the inverse of $f(x)=\dfrac{4x-1}{x+3}$ ($x\neq-3$) and verify it with the input $0$.

11 A cubic inverse ★★★

Show that $f(x)=2x^3-16$ is one-to-one on the real numbers and find $f^{-1}$.

12 Are they inverses? ★★★

Show by composition that $f(x)=2x+3$ and $g(x)=\dfrac{x-3}{2}$ are inverses of each other.

13 Reading a graph and its inverse ★★★

The graph of the one-to-one function $f(x)=\dfrac{x^3}{2}$ is shown below.

(a) Use the points $P$ and $Q$ to find $f^{-1}(4)$ and $f^{-1}(-4)$. (b) Give the coordinates of the two points of the graph of $f^{-1}$ that correspond to $P$ and $Q$. (c) Find a formula for $f^{-1}$.

14 Describing a transformation ★★★

Describe how the graph of $g(x)=-|x-3|+5$ is obtained from $y=|x|$, then give its vertex, its y-intercept, and its x-intercepts.

15 Symmetry proofs ★★★

Prove that $f(x)=\dfrac{x}{x^2+1}$ is odd and that $g(x)=|x|+x^2$ is even. Then show that $h(x)=x^3+x^2$ is neither.

16 Absolute value as pieces ★★★

(a) Rewrite $|3-2x|$ as a piecewise function. (b) Solve $|3-2x|=7$.

17 Converting temperatures ★★★

The Fahrenheit temperature is $F(C)=\dfrac95C+32$, where $C$ is in degrees Celsius. (a) Find the inverse function. (b) A thermometer reads $98.6\,^{\circ}\text{F}$. What is that in Celsius? (c) Explain what $F^{-1}$ represents.

18 Domain traps with a root ★★★

Let $h(x)=\sqrt{x^2-9}$. (a) Decompose $h=f\circ g$. (b) Find the domain of $h$. (c) With your $f$ and $g$, find $(g\circ f)(x)$ and its domain, and explain why it differs from the domain of $h$.

19 Both orders, both domains ★★★

Let $f(x)=\sqrt{4-x}$ and $g(x)=x^2$. Find $f\circ g$ and $g\circ f$ with their domains.

20 Inverse after restriction ★★★

Let $f(x)=x^2+6x+1$ for $x\ge-3$. (a) Complete the square and give the range of $f$. (b) Find $f^{-1}$ with its domain and range. (c) Verify that $(-1,-4)$ is on the graph of $f$ and $(-4,-1)$ is on the graph of $f^{-1}$. (d) What would $f^{-1}$ be if the domain were $x\le-3$?

21 Inverse of a piecewise function ★★★

Let $f(x)=x+4$ if $x\le0$, and $f(x)=2x+6$ if $x\gt0$. (a) Find the range of each piece and explain why $f$ is one-to-one. (b) Find $f^{-1}$ as a piecewise function with its domain. (c) Verify with $f(-3)$ and $f(2)$.

22 Solving with composition ★★★

Let $f(x)=x^2-1$ and $g(x)=2x+3$. (a) Solve $(f\circ g)(x)=24$. (b) Solve $(g\circ f)(x)=x$.

23 Moving points under a transformation ★★★

The points $(-2,3)$, $(0,-1)$ and $(4,5)$ lie on the graph of $y=f(x)$. Find the images of these points on the graph of (a) $y=-2f(x-1)+3$ and (b) $y=\tfrac12 f(-x)$.

24 Parity of composites ★★★

Let $f$ and $g$ be defined on all real numbers. Prove: (a) if $f$ is odd and $g$ is even, then $f\circ g$ is even; (b) if $f$ and $g$ are both odd, then $f\circ g$ is odd. Then illustrate (a) with $f(x)=x^3$ and $g(x)=x^2+1$.