Functions: Domain, Range, Composition and Inverses: Precalculus Math Worksheet – PDF Download

Precalculus Functions Review Worksheet: problems 1–5

Problem 1: Evaluate function notation

Let \(f(x)=3x^{2}-2x+1\). Evaluate each expression.

1. \(f(3)\)
2. \(f(-2)\)
3. \(f(a+1)\)

Problem 2: Domain with square roots

Write the domain of each function in interval notation.

1. \(f(x)=\sqrt{2x-6}\)
2. \(g(x)=\sqrt{10-5x}\)

Problem 3: Domain with denominators

Write the domain of each function in interval notation.

1. \(f(x)=\frac{x+1}{x^{2}-9}\)
2. \(g(x)=\frac{1}{x^{2}+4}\)

Problem 4: Domain with several restrictions

Write the domain of each function in interval notation.

1. \(f(x)=\frac{\sqrt{x+4}}{x-2}\)
2. \(g(x)=\ln(5-x)\)

Problem 5: Domain and range from a graph

The graph shows a function \(f\) with a closed point at \((-1,0)\) and an open point at \((3,0)\).

1. State the domain in interval notation.
2. State the range in interval notation.
3. Find every \(x\) with \(f(x)=3\).

Precalculus Functions Review Worksheet: problems 6–10

Problem 6: Composition from tables

Use the tables to evaluate each expression.

x	1	2	3	4
f(x)	2	4	1	3
g(x)	3	1	4	2

1. \(f(g(1))\)
2. \(g(f(3))\)
3. \(f(f(1))\)
4. \(g(g(4))\)

Problem 7: Composition of polynomials

Let \(f(x)=x^{2}+1\) and \(g(x)=3x-2\).

1. Write and simplify \(f(g(x))\).
2. Write and simplify \(g(f(x))\).
3. Evaluate \(f(g(2))\).

Problem 8: Composition from a mapping diagram

Function \(g\) maps set A to set B and \(f\) maps set B to set C, as shown.

1. Find \(f(g(1))\), \(f(g(2))\) and \(f(g(3))\).
2. Write rules for \(g(x)\) and \(f(x)\), then a formula for \(f(g(x))\).

Problem 9: Domain of a composition (roots)

Let \(f(x)=\sqrt{x}\) and \(g(x)=x-5\). Write each composition and state its domain.

1. \(f(g(x))\)
2. \(g(f(x))\)

Problem 10: Domain of a composition (rational)

Let \(f(x)=\frac{1}{x}\) and \(g(x)=x+2\). Write each composition and state its domain.

1. \(f(g(x))\)
2. \(g(f(x))\)

Precalculus Functions Review Worksheet: problems 11–15

Problem 11: Decompose a function

For each \(h(x)\), write \(f(x)\) and \(g(x)\) (neither equal to \(x\)) so that \(h(x)=f(g(x))\).

1. \(h(x)=(2x+5)^{4}\)
2. \(h(x)=\frac{1}{(x-3)^{2}}\)
3. \(h(x)=\sqrt{x^{2}+9}\)

Problem 12: Inverse of a linear function

Find the inverse function of each.

1. \(f(x)=5x-3\)
2. \(g(x)=-\frac{2}{3}x+4\)

Problem 13: Inverse of a rational function

Let \(f(x)=\frac{2x+1}{x-3}\). Find \(f^{-1}(x)\), then state the domain and range of \(f\).

Problem 14: Verify inverse functions

Decide whether \(f\) and \(g\) are inverse functions by computing both compositions.

1. \(f(x)=x^{3}+4\) and \(g(x)=\sqrt[3]{x-4}\)
2. \(f(x)=\frac{1}{2}x-3\) and \(g(x)=2x+3\)

Problem 15: Inverse values from a graph

The graph shows a one-to-one function \(f\) through the points \((-2,-4)\), \((0,-1)\), \((2,1)\), \((4,3)\) and \((6,7)\).

1. Evaluate \(f(2)\).
2. Evaluate \(f^{-1}(3)\).
3. Evaluate \(f^{-1}(-1)\).
4. Evaluate \(f^{-1}(7)\).

Precalculus Functions Review Worksheet: problems 16–20

Problem 16: Inverse values from tables

Table 1 shows a function \(f\). Table 2 shows a function \(g\).

x	0	1	2	3	4
f(x)	5	3	0	-2	-6

x	1	2	3
g(x)	4	7	4

1. Find \(f^{-1}(0)\) and \(f^{-1}(-2)\).
2. Evaluate \(f^{-1}(f(1))\).
3. Explain why \(g\) has no inverse function.

Problem 17: Horizontal line test

Use the horizontal line test on each graph. State which graphs are functions that have an inverse function.

Problem 18: Restrict a domain: vertex form

The graph shows \(f(x)=(x-3)^{2}\).

1. Explain why \(f\) has no inverse function on all real numbers.
2. Restrict the domain to \(x \ge 3\). Find \(f^{-1}(x)\), with its domain and range.
3. Evaluate \(f^{-1}(16)\).

Problem 19: Restrict a domain: complete the square

Let \(f(x)=x^{2}+4x\) with the domain restricted to \(x \ge -2\). Find \(f^{-1}(x)\) and state its domain and range.

Problem 20: Temperature conversion

The function \(C(F)=\frac{5}{9}(F-32)\) converts degrees Fahrenheit to degrees Celsius.

1. Find the inverse function \(F(C)\).
2. Normal body temperature is \(37^{\circ}\text{C}\). Convert it to Fahrenheit.
3. Show that \(C(F(C))=C\).

Precalculus Functions Review Worksheet: problems 21–24

Problem 21: Coupon and discount order

Maya buys a jacket that costs \(p\) dollars. The store takes 20% off, \(d(p)=0.8p\), and she also has a $5 coupon, \(c(p)=p-5\). The jacket costs $60.

1. Find \(c(d(60))\) and \(d(c(60))\).
2. Write formulas for \(c(d(p))\) and \(d(c(p))\).
3. Explain which order saves more and by how much.

Problem 22: Taxi fare inverse

A taxi charges \(F(m)=2.50+1.80m\) dollars for a ride of \(m\) miles.

1. Find the inverse function \(m(F)\) and explain what it computes.
2. Jamal pays $20.50. How many miles did he ride?
3. Aisha pays $11.50. How many miles did she ride?

Problem 23: Growing ripple

A circular ripple has radius \(r(t)=3t\) feet after \(t\) seconds. Its area is \(A(r)=\pi r^{2}\) square feet.

1. Write \(A(r(t))\) in simplest form.
2. Find the area after 4 seconds, to the nearest tenth.
3. At what time is the area \(81\pi\) square feet?

Problem 24: Inverse of a composition

Let \(f(x)=2x+1\) and \(g(x)=x-4\).

1. Find \(f(g(x))\) and its inverse function.
2. Find \(f^{-1}(x)\) and \(g^{-1}(x)\), then simplify \(g^{-1}(f^{-1}(x))\).
3. Compare your results from (a) and (b).

How to use this precalculus functions review worksheet

Print this precalculus functions review worksheet as a PDF, solve every problem on paper, then compare with the answer key.

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