ACT Math: Find Inverse Functions and Determine Their Domains Systematically

Step 1: Write the function as y=f(x). Example: y=2x+3. Step 2: Swap x and y. Result: x=2y+3. Step 3: Solve for y. x=2y+3 becomes 2y=x-3, so y=(x-3)/2. Step 4: Write the inverse as f⁻¹(x). f⁻¹(x)=(x-3)/2. Step 5: Determine the domain of f⁻¹. The domain of f⁻¹ is the range of f. If f(x)=2x+3 has range all real numbers, then f⁻¹ has domain all real numbers. The key insight is that the domain of the inverse equals the range of the original function.

Example with restriction: f(x)=x² (domain x≥0 to make it one-to-one). The range is y≥0. The inverse is f⁻¹(x)=√x with domain x≥0 (matching the original range). If you forget this restriction, you get f⁻¹(x)=±√x, which is not a function because it produces two outputs for one input.

Mistake 1: Forgetting to swap x and y. You solve y=f(x) for x, which gives the inverse formula, but if you forget the swap, you end up with f(x) again. The swap is essential. Mistake 2: Ignoring domain restrictions. If f(x)=x² is restricted to x≥0, the inverse f⁻¹(x)=√x must have domain x≥0. Forgetting the restriction means f⁻¹ has domain all non-negative reals, not all reals. Always remember: the range of f becomes the domain of f⁻¹. If f has a restricted domain, it affects the range, which affects f⁻¹'s domain.

To verify an inverse, check that f(f⁻¹(x))=x and f⁻¹(f(x))=x. If these do not hold, your inverse is incorrect or your domain restriction is wrong.

Function 1: f(x)=3x-2. Swap: x=3y-2. Solve: y=(x+2)/3. Inverse: f⁻¹(x)=(x+2)/3. Domain: all real numbers. Function 2: f(x)=x² (domain x≥0). Swap: x=y². Solve: y=√x (taking positive root). Inverse: f⁻¹(x)=√x (domain x≥0). Function 3: f(x)=1/x. Swap: x=1/y. Solve: y=1/x. Inverse: f⁻¹(x)=1/x (same function). Domain: x≠0. Function 4: f(x)=2^x. Swap: x=2^y. Solve: y=log₂(x). Inverse: f⁻¹(x)=log₂(x) (domain x>0). For each function, follow the five-step process, identify domain restrictions, and verify your inverse using composition.

After finding the inverse, verify by picking a test value. If f(3)=7 and f⁻¹(7) should equal 3, check that it does. This verification catches errors in algebra or domain reasoning.

Inverse function questions appear 0-2 times per ACT Math section and test your understanding of function behavior, domain, and range. These are not just algebraic manipulation; they require reasoning about what the inverse means and how domain restrictions affect it. Once you master the five-step process and understand the domain-range connection, you solve these questions confidently, earning points on a question type that confuses students who try to memorize patterns.

Spend 20 minutes this week finding inverses for 10 functions (include at least one quadratic and one exponential/logarithmic pair). For each, identify domain restrictions and verify using composition. By test day, finding inverses and determining their domains will be automatic, and you will answer these questions with confidence grounded in understanding, not memorization.