Section 1.inv

Inverse Functions

Inverse functions show up directly on the final — expect 1–2 questions on finding or verifying inverses. They also lay the groundwork for logarithms (inverses of exponentials) and trig inverses you'll see later.

⚡ Quick Summary

An inverse function is like an "undo" button. If a function converts temperatures from Celsius to Fahrenheit, the inverse converts Fahrenheit back to Celsius. Whatever the original function does, the inverse reverses it — you get back exactly where you started.

Inverse Verification

f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x

Both compositions must equal x — checking only one isn't enough

Finding an Inverse Algebraically

y = f(x) \;\longrightarrow\; \text{swap } x \text{ and } y \;\longrightarrow\; \text{solve for } y = f^{-1}(x)

Replace f(x) with y, swap every x and y, then isolate y

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

Not every function can be undone. If two different inputs produce the same output, there's no way to reverse the process because you wouldn't know which input to go back to. A function that *can* be undone is called one-to-one — every output came from exactly one input. The horizontal line test is a quick visual check: if any horizontal line crosses the graph more than once, the function isn't one-to-one.

To actually find an inverse, use the "swap and solve" method: take the equation, swap $x$ and $y$ (because inputs and outputs are switching roles), then solve for $y$ . The graph of the inverse is a mirror image of the original, reflected over the diagonal line $y = x$ .

To verify two functions are inverses, plug one into the other in both directions. If $f(g(x)) = x$ *and* $g(f(x)) = x$ , they perfectly undo each other. You need to check both directions — one alone isn't enough.

Key Formulas

Inverse Verification

f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x

Both compositions must equal x — checking only one isn't enough

Finding an Inverse Algebraically

y = f(x) \;\longrightarrow\; \text{swap } x \text{ and } y \;\longrightarrow\; \text{solve for } y = f^{-1}(x)

Replace f(x) with y, swap every x and y, then isolate y

Domain–Range Relationship

\text{Domain of } f = \text{Range of } f^{-1}, \quad \text{Range of } f = \text{Domain of } f^{-1}

Inputs and outputs swap roles when you invert

Key Takeaways

✓To find $f^{-1}$ : replace $f(x)$ with $y$ , swap $x$ and $y$ , then solve for $y$
✓A function has an inverse only if it's one-to-one — use the horizontal line test to check
✓Domain of $f$ = Range of $f^{-1}$ , and Range of $f$ = Domain of $f^{-1}$
✓To verify inverses: both $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ must hold

⚠️ Common Mistakes

✗Solving for $x$ without swapping first — if you skip the swap step, you'll just get back the original function instead of the inverse
✗Confusing $f^{-1}(x)$ with $\frac{1}{f(x)}$ — the $-1$ is NOT an exponent; $f^{-1}$ means the inverse function, not the reciprocal
✗Checking only one composition direction when verifying — you need BOTH $f(f^{-1}(x)) = x$ AND $f^{-1}(f(x)) = x$
✗Taking the wrong square root when the domain is restricted — if the original has $x \geq 0$ , the inverse must use the positive root only

Builds on: 1.comp

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1.comp Composition & Transformations

2.lin Linear Functions