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Inverse functions practice problems

Find the inverse of a one-to-one function, and check it by composing. 60 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.

16 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 13.34

medium
Find the inverse of \( \displaystyle f(x) = \frac{4 - 3 x}{x - 4} \).
  1. Write y = f(x) and solve for x. Clear the denominator, collect the x terms, and factor x out. Then rename y as x.
    Reviewed
  2. \[ \left. \frac{4 - 3 x}{x - 4} \right|_{\substack{ x=\frac{4 \left(x + 1\right)}{x + 3} }} = x \]
    f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
  3. \[ \left. \frac{4 \left(x + 1\right)}{x + 3} \right|_{\substack{ x=\left. \frac{4 - 3 x}{x - 4} \right|_{\substack{ x=2 }} }} = 2 \]
    And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = \frac{4 \left(x + 1\right)}{x + 3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.37

medium
Find the inverse of \( \displaystyle f(x) = \ln{\left(x - 3 \right)} + 2 \).
  1. Write y = f(x) and solve for x. Isolate the log, then exponentiate. Then rename y as x.
    Reviewed
  2. \[ \left. \ln{\left(x - 3 \right)} + 2 \right|_{\substack{ x=e^{x - 2} + 3 }} = x \]
    f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
  3. \[ \left. e^{x - 2} + 3 \right|_{\substack{ x=\left. \ln{\left(x - 3 \right)} + 2 \right|_{\substack{ x=5 }} }} = 5 \]
    And f⁻¹(f(5)) = 5.✓ Proved
Answer \( f^{-1}(x) = e^{x - 2} + 3 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.38

medium
Find the inverse of \( \displaystyle f(x) = -3 + e^{- x} \).
  1. Write y = f(x) and solve for x. Isolate the exponential, then take the natural log. Then rename y as x.
    Reviewed
  2. \[ \left. -3 + e^{- x} \right|_{\substack{ x=- \ln{\left(x + 3 \right)} }} = x \]
    f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
  3. \[ \left. - \ln{\left(x + 3 \right)} \right|_{\substack{ x=\left. -3 + e^{- x} \right|_{\substack{ x=2 }} }} = 2 \]
    And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = - \ln{\left(x + 3 \right)} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.39

medium
Find the inverse of \( \displaystyle f(x) = e^{- 2 x} \).
  1. Write y = f(x) and solve for x. Isolate the exponential, then take the natural log. Then rename y as x.
    Reviewed
  2. \[ \left. e^{- 2 x} \right|_{\substack{ x=\frac{\left(-1\right) \ln{\left(x \right)}}{2} }} = x \]
    f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
  3. \[ \left. \frac{\left(-1\right) \ln{\left(x \right)}}{2} \right|_{\substack{ x=\left. e^{- 2 x} \right|_{\substack{ x=2 }} }} = 2 \]
    And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = - \frac{\ln{\left(x \right)}}{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.42

medium
Find the inverse of \( \displaystyle f(x) = -4 + e^{- 2 x} \).
  1. Write y = f(x) and solve for x. Isolate the exponential, then take the natural log. Then rename y as x.
    Reviewed
  2. \[ \left. -4 + e^{- 2 x} \right|_{\substack{ x=\frac{\left(-1\right) \ln{\left(x + 4 \right)}}{2} }} = x \]
    f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
  3. \[ \left. \frac{\left(-1\right) \ln{\left(x + 4 \right)}}{2} \right|_{\substack{ x=\left. -4 + e^{- 2 x} \right|_{\substack{ x=2 }} }} = 2 \]
    And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = - \frac{\ln{\left(x + 4 \right)}}{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.163

medium
Find the inverse of \( \displaystyle f(x) = e^{4 x} - 1 \).
  1. Write y = f(x) and solve for x. Isolate the exponential, then take the natural log. Then rename y as x.
    Reviewed
  2. \[ \left. e^{4 x} - 1 \right|_{\substack{ x=\frac{\ln{\left(x + 1 \right)}}{4} }} = x \]
    f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
  3. \[ \left. \frac{\ln{\left(x + 1 \right)}}{4} \right|_{\substack{ x=\left. e^{4 x} - 1 \right|_{\substack{ x=2 }} }} = 2 \]
    And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = \frac{\ln{\left(x + 1 \right)}}{4} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.165

medium
Find the inverse of \( \displaystyle f(x) = \ln{\left(x - 1 \right)} - 1 \).
  1. Write y = f(x) and solve for x. Isolate the log, then exponentiate. Then rename y as x.
    Reviewed
  2. \[ \left. \ln{\left(x - 1 \right)} - 1 \right|_{\substack{ x=e^{x + 1} + 1 }} = x \]
    f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
  3. \[ \left. e^{x + 1} + 1 \right|_{\substack{ x=\left. \ln{\left(x - 1 \right)} - 1 \right|_{\substack{ x=3 }} }} = 3 \]
    And f⁻¹(f(3)) = 3.✓ Proved
Answer \( f^{-1}(x) = e^{x + 1} + 1 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.171

medium
Find the inverse of \( \displaystyle f(x) = \frac{5 x - 5}{- x - 4} \).
  1. Write y = f(x) and solve for x. Clear the denominator, collect the x terms, and factor x out. Then rename y as x.
    Reviewed
  2. \[ \left. \frac{5 x - 5}{- x - 4} \right|_{\substack{ x=\frac{5 - 4 x}{x + 5} }} = x \]
    f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
  3. \[ \left. \frac{5 - 4 x}{x + 5} \right|_{\substack{ x=\left. \frac{5 x - 5}{- x - 4} \right|_{\substack{ x=2 }} }} = 2 \]
    And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = \frac{5 - 4 x}{x + 5} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.221

medium
Find the inverse of \( \displaystyle f(x) = \frac{- 4 x - 2}{- 3 x - 4} \).
  1. Write y = f(x) and solve for x. Clear the denominator, collect the x terms, and factor x out. Then rename y as x.
    Reviewed
  2. \[ \left. \frac{- 4 x - 2}{- 3 x - 4} \right|_{\substack{ x=\frac{2 \left(1 - 2 x\right)}{3 x - 4} }} = x \]
    f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
  3. \[ \left. \frac{2 \left(1 - 2 x\right)}{3 x - 4} \right|_{\substack{ x=\left. \frac{- 4 x - 2}{- 3 x - 4} \right|_{\substack{ x=2 }} }} = 2 \]
    And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = \frac{2 \left(1 - 2 x\right)}{3 x - 4} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.233

medium
Find the inverse of \( \displaystyle f(x) = e^{4 x} \).
  1. Write y = f(x) and solve for x. Isolate the exponential, then take the natural log. Then rename y as x.
    Reviewed
  2. \[ \left. e^{4 x} \right|_{\substack{ x=\frac{\ln{\left(x \right)}}{4} }} = x \]
    f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
  3. \[ \left. \frac{\ln{\left(x \right)}}{4} \right|_{\substack{ x=\left. e^{4 x} \right|_{\substack{ x=2 }} }} = 2 \]
    And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = \frac{\ln{\left(x \right)}}{4} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.234

medium
Find the inverse of \( \displaystyle f(x) = \frac{5 x - 1}{2 x + 2} \).
  1. Write y = f(x) and solve for x. Clear the denominator, collect the x terms, and factor x out. Then rename y as x.
    Reviewed
  2. \[ \left. \frac{5 x - 1}{2 x + 2} \right|_{\substack{ x=\frac{- 2 x - 1}{2 x - 5} }} = x \]
    f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
  3. \[ \left. \frac{- 2 x - 1}{2 x - 5} \right|_{\substack{ x=\left. \frac{5 x - 1}{2 x + 2} \right|_{\substack{ x=2 }} }} = 2 \]
    And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = \frac{- 2 x - 1}{2 x - 5} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.379

medium
Find the inverse of \( \displaystyle f(x) = \ln{\left(x \right)} + 1 \).
  1. Write y = f(x) and solve for x. Isolate the log, then exponentiate. Then rename y as x.
    Reviewed
  2. \[ \left. \ln{\left(x \right)} + 1 \right|_{\substack{ x=e^{x - 1} }} = x \]
    f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
  3. \[ \left. e^{x - 1} \right|_{\substack{ x=\left. \ln{\left(x \right)} + 1 \right|_{\substack{ x=1 }} }} = 1 \]
    And f⁻¹(f(1)) = 1.✓ Proved
Answer \( f^{-1}(x) = e^{x - 1} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.32

easy
Find the inverse of \( \displaystyle f(x) = 2 - 4 x \).
  1. Write y = f(x) and solve for x. Subtract the constant and divide. Then rename y as x.
    Reviewed
  2. \[ \left. 2 - 4 x \right|_{\substack{ x=- \frac{x}{4} + 1 \cdot \frac{1}{2} }} = x \]
    f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
  3. \[ \left. - \frac{x}{4} + 1 \cdot \frac{1}{2} \right|_{\substack{ x=\left. 2 - 4 x \right|_{\substack{ x=1 }} }} = 1 \]
    And f⁻¹(f(1)) = 1.✓ Proved
Answer \( f^{-1}(x) = \frac{1}{2} - \frac{x}{4} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.33

easy
Find the inverse of \( \displaystyle f(x) = 3 x - 1 \).
  1. Write y = f(x) and solve for x. Subtract the constant and divide. Then rename y as x.
    Reviewed
  2. \[ \left. 3 x - 1 \right|_{\substack{ x=\frac{x}{3} + 1 \cdot \frac{1}{3} }} = x \]
    f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
  3. \[ \left. \frac{x}{3} + 1 \cdot \frac{1}{3} \right|_{\substack{ x=\left. 3 x - 1 \right|_{\substack{ x=2 }} }} = 2 \]
    And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = \frac{x}{3} + \frac{1}{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.35

easy
Find the inverse of \( \displaystyle f(x) = 4 x + 4 \).
  1. Write y = f(x) and solve for x. Subtract the constant and divide. Then rename y as x.
    Reviewed
  2. \[ \left. 4 x + 4 \right|_{\substack{ x=\frac{x}{4} - 1 }} = x \]
    f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
  3. \[ \left. \frac{x}{4} - 1 \right|_{\substack{ x=\left. 4 x + 4 \right|_{\substack{ x=2 }} }} = 2 \]
    And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = \frac{x}{4} - 1 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.223

easy
Find the inverse of \( \displaystyle f(x) = - 5 x - 5 \).
  1. Write y = f(x) and solve for x. Subtract the constant and divide. Then rename y as x.
    Reviewed
  2. \[ \left. - 5 x - 5 \right|_{\substack{ x=\frac{\left(-1\right) x}{5} - 1 }} = x \]
    f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
  3. \[ \left. \frac{\left(-1\right) x}{5} - 1 \right|_{\substack{ x=\left. - 5 x - 5 \right|_{\substack{ x=2 }} }} = 2 \]
    And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = - \frac{x}{5} - 1 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More inverse functions practice

Find the inverse of \( \displaystyle f(x) = - x - 3 \).
Problem 13.36easy✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = 5 x - 6 \).
Problem 13.40easy✓ Nihil obstat
Find the inverse of \( \displaystyle f(x) = - 4 x - 5 \).
Problem 13.41easy✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = - 5 x - 3 \).
Problem 13.43easy✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = 4 x + 6 \).
Problem 13.168easy✓ Nihil obstat
Find the inverse of \( \displaystyle f(x) = 2 x + 5 \).
Problem 13.170easy✓ Nihil obstat
Find the inverse of \( \displaystyle f(x) = - 3 x - 4 \).
Problem 13.224easy✓ Nihil obstat
Find the inverse of \( \displaystyle f(x) = 6 - x \).
Problem 13.225easy✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = 5 x + 6 \).
Problem 13.229easy✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = 4 x + 2 \).
Problem 13.230easy✓ Nihil obstat
Find the inverse of \( \displaystyle f(x) = 4 x - 3 \).
Problem 13.371easy✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = 3 - x \).
Problem 13.378easy✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = 5 x - 1 \).
Problem 13.380easy✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = 2 x + 2 \).
Problem 13.449easy✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = 2 x - 4 \).
Problem 13.454easy✓ Nihil obstat
Find the inverse of \( \displaystyle f(x) = - 5 x - 1 \).
Problem 13.456easy✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = e^{- 4 x} \).
Problem 13.31medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = \ln{\left(x - 1 \right)} + 1 \).
Problem 13.44medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = \frac{x + 3}{4 - x} \).
Problem 13.45medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = \ln{\left(x - 4 \right)} + 1 \).
Problem 13.164medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = \ln{\left(x - 5 \right)} + 2 \).
Problem 13.166medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = \ln{\left(x - 5 \right)} + 3 \).
Problem 13.167medium✓ Nihil obstat
Find the inverse of \( \displaystyle f(x) = \frac{5 x + 2}{- x - 3} \).
Problem 13.169medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = \ln{\left(x - 1 \right)} + 3 \).
Problem 13.172medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = -6 + e^{- 3 x} \).
Problem 13.220medium✓ Nihil obstat
Find the inverse of \( \displaystyle f(x) = \ln{\left(x - 5 \right)} - 2 \).
Problem 13.222medium✓ Nihil obstat
Find the inverse of \( \displaystyle f(x) = \frac{4 - 2 x}{x + 4} \).
Problem 13.226medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = \frac{x + 6}{2 x} \).
Problem 13.227medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = \ln{\left(x - 3 \right)} - 2 \).
Problem 13.228medium✓ Nihil obstat
Find the inverse of \( \displaystyle f(x) = \ln{\left(x - 6 \right)} + 1 \).
Problem 13.231medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = e^{3 x} + 3 \).
Problem 13.232medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = \frac{- 3 x - 1}{- x - 1} \).
Problem 13.372medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = \ln{\left(x - 4 \right)} - 2 \).
Problem 13.373medium✓ Nihil obstat
Find the inverse of \( \displaystyle f(x) = \frac{- 5 x - 5}{- 2 x - 1} \).
Problem 13.374medium✓ Nihil obstat
Find the inverse of \( \displaystyle f(x) = \frac{- x - 2}{4 - 2 x} \).
Problem 13.375medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = \ln{\left(x - 4 \right)} + 3 \).
Problem 13.376medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = 1 + e^{- x} \).
Problem 13.377medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = -5 + e^{- 4 x} \).
Problem 13.450medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = \ln{\left(x - 6 \right)} - 1 \).
Problem 13.451medium✓ Nihil obstat
Find the inverse of \( \displaystyle f(x) = e^{4 x} + 4 \).
Problem 13.452medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = e^{4 x} - 6 \).
Problem 13.453medium✓ Nihil obstat
Find the inverse of \( \displaystyle f(x) = \frac{2 x + 5}{3 x - 1} \).
Problem 13.455medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = 2 + e^{- 2 x} \).
Problem 13.457medium✓ Every equation proved
Find the inverse of \( \displaystyle f(x) = \frac{5 x + 4}{2 x - 3} \).
Problem 13.458medium✓ Every equation proved