Inverse functions
Find the inverse of a function and verify inverse pairs.
Objective
Find the inverse of a function and verify inverse pairs.
What it means
The inverse of a function f, written f⁻¹, reverses the input-output pairs of f. To find it algebraically, swap x and y in the equation and solve for the new y.
Key terms
- Inverse function
- A function that undoes another function: f(f⁻¹(x)) = x.
- One-to-one
- A function where every output comes from exactly one input, required for an inverse to be a function.
Rules to remember
- Replace f(x) with y.
- Swap x and y.
- Solve the new equation for y.
- Replace y with f⁻¹(x).
Step by step
- 1Write y = f(x)Start with the function in y = form.
- 2Swap x and yExchange the two variables.
- 3Solve for yIsolate y using algebra.
- 4RenameWrite the result as f⁻¹(x).
Common mistakes
- Forgetting to swap x and y before solving.
- Mixing up f⁻¹(x) with 1/f(x), which are not the same.
Tips
- Check your answer: f(f⁻¹(x)) should simplify to x.
- Inverses of lines are lines; the slope becomes its reciprocal-related value.