Reverse a Linear Function’s Operations
A nonconstant linear function is one-to-one.
About 25 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: Find the inverse of f(x)=3x+2. Candidate answers: (A) f⁻¹(x)=(x+2)/3 (B) f⁻¹(x)=(x−2)/3 (C) f⁻¹(x)=1/(3x+2) (D) f⁻¹(x)=3x−2.
- Independent check inputs: Find the inverse of f(x)=3x+3. Candidate answers: (A) f⁻¹(x)=(x+3)/3 (B) f⁻¹(x)=(x−3)/3 (C) f⁻¹(x)=1/(3x+3) (D) f⁻¹(x)=3x−3.
Learn the idea
Function operations must respect input order and domain. A composition feeds one output into another function, while an inverse reverses a one-to-one relationship and exchanges the original domain and range. A nonconstant linear function is one-to-one. Solving its output equation for the original input undoes its addition and multiplication in reverse order, producing the inverse function.
Here is the method on the supplied teaching case. Set y=3x+2, then solve x=(y−2)/3. Swap the input/output labels. An inverse undoes a function; it is not generally its reciprocal.
An inverse-function symbol is not a reciprocal exponent. Check by composing in both orders; a valid inverse returns the starting input, subject to the functions’ stated domains.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: Find the inverse of f(x)=3x+2. Candidate answers: (A) f⁻¹(x)=(x+2)/3 (B) f⁻¹(x)=(x−2)/3 (C) f⁻¹(x)=1/(3x+2) (D) f⁻¹(x)=3x−2. Solution: Set y=3x+2, then solve x=(y−2)/3. Swap the input/output labels. An inverse undoes a function; it is not generally its reciprocal. The final answer is f⁻¹(x)=(x−2)/3. An inverse-function symbol is not a reciprocal exponent. Check by composing in both orders; a valid inverse returns the starting input, subject to the functions’ stated domains.
Check your reasoning
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Helpful tips
- A nonconstant linear function is one-to-one.
- An inverse-function symbol is not a reciprocal exponent.
See the answer and explanation
f⁻¹(x)=(x−3)/3
Set y=3x+3, then solve x=(y−3)/3. Swap the input/output labels. An inverse undoes a function; it is not generally its reciprocal.
Take it further
Changed-input task: Find the inverse of f(x)=3x+9. Candidate answers: (A) f⁻¹(x)=(x+9)/3 (B) f⁻¹(x)=(x−9)/3 (C) f⁻¹(x)=1/(3x+9) (D) f⁻¹(x)=3x−9. Show the mathematical reason, not only the selected answer. Expected result: f⁻¹(x)=(x−9)/3. Set y=3x+9, then solve x=(y−9)/3. Swap the input/output labels. An inverse undoes a function; it is not generally its reciprocal.
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