ACT Preparation / Unit 10 ACT 0926 worked example, 60 to 90 seconds
Finding and Checking an Inverse
The idea
An inverse is found and checked in four steps. First, confirm the function is one-to-one, restricting its domain if needed. Second, write y = f(x) and swap x and y. Third, solve for y and rename it f⁻¹(x). Fourth, check by composing both ways: f(f⁻¹(x)) and f⁻¹(f(x)) must each simplify to x.
In practice
Worked example: f(x) = (x + 5)/3. Step two: x = (y + 5)/3. Step three: 3x = y + 5, so f⁻¹(x) = 3x − 5. Step four: f(f⁻¹(x)) = ((3x − 5) + 5)/3 = x, and f⁻¹(f(x)) = 3((x + 5)/3) − 5 = x. Both simplify to x, so the inverse is right.
Where people slip
A function that is not one-to-one, such as x², has no inverse until its domain is restricted, for example to x ≥ 0, where the inverse is √x.
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