Common Core Math
Sign in
Back to Solve for inverse function

Exercises: Solving for Inverse Functions

Show all algebraic steps for each problem. Write inverse functions using $f^{-1}(x)$ notation.

Grade 9·21 problems·~30 min·Common Core Math - HS Functions·standard·hsf-bf-b-4a
Work through problems with immediate feedback
A

Warm-Up: Review What You Know

These problems review skills you have already learned.

1.

If ff and gg are inverse functions, which of the following must be true?

2.

Solve the equation 3x−7=113x - 7 = 11 for xx.

3.

For f(x)=2x+5f(x) = 2x + 5, what is the value of xx when f(x)=13f(x) = 13?

B

Fluency Practice

Find the inverse of each function. Show your steps.

1.

Find f−1(x)f^{-1}(x) for f(x)=4x−3f(x) = 4x - 3.

Express your answer in the form f−1(x)=x+abf^{-1}(x) = \frac{x + a}{b}.
Enter just the value of aa (the number added in the numerator).

2.

Find f−1(x)f^{-1}(x) for f(x)=x−52f(x) = \dfrac{x - 5}{2}.

Enter the coefficient of xx in the expression for f−1(x)f^{-1}(x).
(For example, if f−1(x)=2x+5f^{-1}(x) = 2x + 5, enter 2.)

Four-step flowchart of the swap-and-solve procedure applied to f(x) = 2x³, showing each algebraic transformation
3.

Find f−1(x)f^{-1}(x) for f(x)=2x3f(x) = 2x^3.

Enter the denominator of the expression under the cube root in f−1(x)f^{-1}(x).
(For example, if f−1(x)=xk3f^{-1}(x) = \sqrt[3]{\frac{x}{k}}, enter kk.)

4.

Let f(x)=x2f(x) = x^2 with domain x≥0x \geq 0.
Which of the following is f−1(x)f^{-1}(x)?

5.

Find f−1(x)f^{-1}(x) for f(x)=x+3x−2f(x) = \dfrac{x + 3}{x - 2}, where x≠2x \neq 2.

After applying the swap-and-solve method, the inverse is of the form
f−1(x)=ax+bx−cf^{-1}(x) = \dfrac{ax + b}{x - c}.
Enter the value of cc.

C

Varied Practice

1.

A student starts finding f−1(x)f^{-1}(x) for f(x)=7x−2f(x) = 7x - 2 and writes:

"Step 1: y=7x−2y = 7x - 2. Step 2: Solve for xx: x=y+27x = \dfrac{y + 2}{7}.
Therefore f−1(x)=y+27f^{-1}(x) = \dfrac{y + 2}{7}."

The student's answer is incomplete. What must they do next?

2.

Complete the swap-and-solve steps for f(x)=5x+9f(x) = 5x + 9.

Step 1: Write y=5x+9y = 5x + 9.
Step 2: Swap xx and yy: x=‾y+‾x = \underline{\hspace{5em}}y + \underline{\hspace{5em}}.
Step 3: Subtract   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   from both sides: x−9=5yx - 9 = 5y.
Step 4: Divide both sides by   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   : f−1(x)=x−9000000f^{-1}(x) = \frac{x-9}{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}}.

coefficient of y (Step 2):
constant (Step 2):
value subtracted (Step 3):
divisor (Step 4):
denominator (Step 4):
3.

Which of the following functions does NOT have an inverse on its natural domain (all real numbers)?

4.

Consider f(x)=x+1x−1f(x) = \dfrac{x + 1}{x - 1}, x≠1x \neq 1 — the standard example from HSF.BF.B.4.a.

A student finds the inverse and gets f−1(x)=x+1x−1f^{-1}(x) = \dfrac{x + 1}{x - 1}.
The student says, "My answer must be wrong — the inverse can't equal the original function."

Is the student correct?

5.

Find f−1(x)f^{-1}(x) for f(x)=3x+1f(x) = 3x + 1. Then verify your answer by computing f ⁣(f−1(x))f\!\left(f^{-1}(x)\right) and showing it equals xx.

D

Word Problems

1.

A bakery uses the formula C(n)=0.75n+4C(n) = 0.75n + 4 to calculate the total cost CC (in dollars) of ordering nn custom cookies.

1.

Solve C(n)=16C(n) = 16 to find how many cookies can be ordered for exactly $16.

2.

Find the inverse function C−1(x)C^{-1}(x) and use it to verify your answer from part (a).
Enter the coefficient of xx in the expression for C−1(x)C^{-1}(x).
(For example, if C−1(x)=kx−mC^{-1}(x) = kx - m, enter kk.)

2.

A sculptor uses the formula V(r)=2r3V(r) = 2r^3 to calculate the volume VV (in cubic inches) of a spherical clay bead with radius rr (in inches). She wants a bead with volume 54 cubic inches.

Find rr by using the inverse of VV. Enter your answer as a whole number.

3.

A student claims that the function T(x)=−2x+6T(x) = -2x + 6 converts temperature in one scale to another.

Find T−1(x)T^{-1}(x) and use it to determine which input xx gives T(x)=0T(x) = 0.
Enter the value of xx as a whole number.

E

Error Analysis

Each problem shows a student's work that contains an error. Identify and explain the mistake.

1.

Maya found the inverse of f(x)=8x−5f(x) = 8x - 5:

  1. Write y=8x−5y = 8x - 5.
  2. Solve for xx: x=y+58x = \dfrac{y + 5}{8}.
  3. Therefore f−1(x)=y+58f^{-1}(x) = \dfrac{y + 5}{8}.

What error did Maya make?

2.

Tyler found the inverse of g(x)=x+2x−3g(x) = \dfrac{x + 2}{x - 3}:

  1. Swap: x=y+2y−3x = \dfrac{y + 2}{y - 3}.
  2. Cross-multiply: x(y−3)=y+2x(y - 3) = y + 2.
  3. Expand: xy−3x=y+2xy - 3x = y + 2.
  4. Collect yy: xy−y=2+3xxy - y = 2 + 3x.
  5. Hmm, I messed up the sign — let me just write y=3x+2x−1y = \dfrac{3x + 2}{x - 1}.

Tyler made a sign error in step 4. What should the right side of step 4 be?

F

Challenge / Extension

These problems require multi-step reasoning. They are bonus problems.

1.

Let f(x)=2x−1x+4f(x) = \dfrac{2x - 1}{x + 4}.

Find f−1(x)f^{-1}(x). The inverse is of the form f−1(x)=ax+bc−xf^{-1}(x) = \dfrac{ax + b}{c - x}.
Enter the value of bb.

2.

Explain why f(x)=x2−3f(x) = x^2 - 3 does not have an inverse on its natural domain, but does have an inverse when the domain is restricted to x≥0x \geq 0. In your response:

  1. State why ff fails the one-to-one requirement on all reals (give a specific example).
  2. Find f−1(x)f^{-1}(x) on the restricted domain x≥0x \geq 0.
  3. Verify your inverse by showing f(f−1(x))=xf(f^{-1}(x)) = x.
0 of 21 answered