M151Q Precalculus · Unit 2: Powers, exponentials, logarithms · on Exam 2
MFG 5.1 Inverse functions
Study notes for Inverse functions in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- The inverse $f^{-1}$ undoes $f$: if $f(a) = b$ then $f^{-1}(b) = a$. Inputs and outputs swap, so the domain of $f^{-1}$ is the range of $f$. The graph of $f^{-1}$ is the reflection of $f$ across $y = x$.
- Find a formula: write $y = f(x)$, swap $x$ and $y$, solve for $y$. Check with $f(f^{-1}(x)) = x$.
- Only one-to-one functions (horizontal line test) have inverses. $x^2$ fails; restrict to $x \ge 0$ and the inverse is $\sqrt x$.
- $f^{-1}(x)$ is not $1/f(x)$.
Key formulas
Inverse relationship
$$f(a) = b \iff f^{-1}(b) = a,\qquad f(f^{-1}(x)) = x,\quad f^{-1}(f(x)) = x$$
Finding the formula
$$y = f(x)\ \xrightarrow{\ \text{swap}\ }\ x = f(y)\ \xrightarrow{\ \text{solve}\ }\ y = f^{-1}(x)$$
Worked example
Find $f^{-1}(x)$ for $f(x) = \dfrac{2x - 1}{3}$ and state the value of $f^{-1}(5)$.
Show the solution
$x = \dfrac{2y-1}{3} \Rightarrow 3x = 2y - 1 \Rightarrow y = \dfrac{3x+1}{2}$. $f^{-1}(5) = 8$; check $f(8) = 15/3 = 5$.
Common mistakes
- Writing 1/f(x) for the inverse.
- Forgetting to swap x and y before solving.
- Claiming every function has an inverse.
Exam tipTo read f⁻¹(b) from a table or graph of f, find where the output is b and report the input.
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