The inverse "undoes" what the function does.

Finding an Inverse

Steps:

1. Write $y = f(x)$
2. Swap $x$ and $y$
3. Solve for $y$
4. Replace $y$ with $f^{-1}(x)$

Example: Find inverse of $f(x) = 2x + 3$

1. $y = 2x + 3$
2. $x = 2y + 3$
3. $x - 3 = 2y$, so $y = \frac{x - 3}{2}$
4. $f^{-1}(x) = \frac{x - 3}{2}$

Domain and Range

• Domain of $f$ = Range of $f^{-1}$
• Range of $f$ = Domain of $f^{-1}$

Graphing

The graph of $f^{-1}$ is the reflection of $f$ over the line $y = x$.

Horizontal Line Test

$f$ has an inverse function if and only if no horizontal line intersects the graph more than once.

Verifying

To verify $g$ is the inverse of $f$:

Check that $f(g(x)) = x$ AND $g(f(x)) = x$

Step 2: Swap $x$ and $y$ $x = y + 7$

Step 3: Solve for $y$ $y = x - 7$

Step 4: Write inverse $f^{-1}(x) = x - 7$

Answer: $f^{-1}(x) = x - 7$

Step 3: Solve for $y$ $3x = y - 1$ $y = 3x + 1$

Step 4: Write inverse $f^{-1}(x) = 3x + 1$

Verify: $f(f^{-1}(x)) = \frac{(3x + 1) - 1}{3} = \frac{3x}{3} = x$ ✓

Answer: $f^{-1}(x) = 3x + 1$

Step 3: Solve for $y$ (multiply both sides by denominator) $x(y - 1) = 2y + 3$ $xy - x = 2y + 3$

Group $y$ terms: $xy - 2y = x + 3$ $y(x - 2) = x + 3$ $y = \frac{x + 3}{x - 2}$

Answer: $f^{-1}(x) = \frac{x + 3}{x - 2}$