How to Find an Inverse Function and Check Both Compositions

To find an inverse, write y = f(x), solve for x, and exchange the variable names. Then check both f(f⁻¹(x)) = x and f⁻¹(f(x)) = x on their proper domains.

First ask whether an inverse function exists

An inverse reverses a function’s input-output assignment. For that reversal to remain a function, each permitted output must come from only one input. This is the one-to-one requirement. A formula alone is not enough: its domain can determine whether the requirement holds.

For example, x² on all real numbers is not one-to-one because 2 and −2 both produce 4. Restricting its domain to x ≥ 0 gives an inverse √x. Restricting instead to x ≤ 0 gives −√x. State the restriction before choosing a square-root branch.

Worked example: reverse a linear rule

Let f(x) = (3x − 5)/2, with all real inputs. Write y = (3x − 5)/2. Multiply by 2 to get 2y = 3x − 5, add 5, and divide by 3: x = (2y + 5)/3. Therefore f⁻¹(x) = (2x + 5)/3.

Check the first composition: f(f⁻¹(x)) = [3(2x + 5)/3 − 5]/2 = (2x + 5 − 5)/2 = x. Check the second: f⁻¹(f(x)) = [2(3x − 5)/2 + 5]/3 = (3x − 5 + 5)/3 = x. Both are valid for every real x.

Domain and range exchange roles

If f maps domain D onto its range R one-to-one, then f⁻¹ takes inputs from R and returns values in D. For f(x) = √(x − 4), the original domain is [4, ∞) and range is [0, ∞). Its inverse is x² + 4 with input restricted to x ≥ 0.

Without that input restriction, x² + 4 would assign outputs to negative inputs that never belonged to the original range. Algebraic manipulation must therefore travel with the domain information, rather than replacing it.

Common mistake

The notation f⁻¹ is not the reciprocal 1/f. For f(x) = 2x + 3, the inverse is (x − 3)/2, whereas the reciprocal is 1/(2x + 3). They perform different operations. Also, one successful numerical example does not establish an inverse identity; simplify both compositions algebraically and check where each is defined.

Yaad rakho: inverse undo karta hai—use reciprocal mat samjho.

Three practice questions with worked answers

1. Find the inverse of f(x) = 5x + 2 over the reals.

   Solve y = 5x + 2 for x: x = (y − 2)/5. Thus f⁻¹(x) = (x − 2)/5. Substituting either way returns the original input.

2. Find the inverse of f(x) = x³ − 7 over the reals.

   From y + 7 = x³, take the real cube root. The inverse is ∛(x + 7). Cubing is one-to-one over all real numbers.

3. Find the inverse of f(x) = x² for x ≤ 0.

   The range is [0, ∞). Use f⁻¹(x) = −√x with x ≥ 0, since the inverse must return nonpositive original inputs.

Related study guides

• Domain and Range from a Function Rule
• Sets, Relations and Functions Class 11 Notes with Examples

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References and authorship

Original examples and practice questions, prepared with AI assistance and editorial arithmetic review. Related definitions and methods: OpenStax: Inverse Functions. No textbook or exam-board affiliation is claimed.