Inverse Function Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

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Find the inverse of f(x)=2x+5xโˆ’1f(x) = \frac{2x + 5}{x - 1} for xโ‰ 1x \neq 1.

Solution

  1. 1
    Set y=2x+5xโˆ’1y = \frac{2x + 5}{x - 1} and swap: x=2y+5yโˆ’1x = \frac{2y + 5}{y - 1}.
  2. 2
    Multiply both sides by (yโˆ’1)(y - 1): x(yโˆ’1)=2y+5x(y - 1) = 2y + 5.
  3. 3
    Expand: xyโˆ’x=2y+5xy - x = 2y + 5. Group yy terms: xyโˆ’2y=x+5xy - 2y = x + 5.
  4. 4
    Factor: y(xโˆ’2)=x+5y(x - 2) = x + 5, so y=x+5xโˆ’2y = \frac{x + 5}{x - 2}.

Answer

fโˆ’1(x)=x+5xโˆ’2f^{-1}(x) = \frac{x + 5}{x - 2}
Rational functions often have inverses that are also rational. The swap-and-solve method works the same way โ€” the algebra just involves clearing fractions and collecting terms with the new variable.

About Inverse Function

The inverse of a function ff is a function fโˆ’1f^{-1} that reverses ff: if f(a)=bf(a) = b then fโˆ’1(b)=af^{-1}(b) = a. It exists only when ff is one-to-one.

Learn more about Inverse Function โ†’

More Inverse Function Examples

Example 1 easy

Find the inverse of [formula].

Example 3 easy

Find the inverse of [formula].

Example 4 hard

If [formula], find [formula]. Then verify by showing that [formula].

โ† All Inverse Function Examples Inverse Function Concept