Inverse Function Formula

Inverse function is the inverse of a function f is a function f^-1 that reverses f: if f(a) = b then f^-1(b) = a.

The Formula

fโˆ’1(f(x))=xf^{-1}(f(x)) = x and f(fโˆ’1(x))=xf(f^{-1}(x)) = x

When to use: If ff turns aa into bb, then fโˆ’1f^{-1} turns bb back into aa. Reverse the process.

Quick Example

f(x)=2xf(x) = 2x (double).
fโˆ’1(x)=x2f^{-1}(x) = \frac{x}{2} (halve).
fโˆ’1(f(3))=fโˆ’1(6)=3f^{-1}(f(3)) = f^{-1}(6) = 3.

Notation

fโˆ’1f^{-1} denotes the inverse function. To find it: write y=f(x)y = f(x), swap xx and yy, solve for yy.

What This Formula Means

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.

If ff turns aa into bb, then fโˆ’1f^{-1} turns bb back into aa. Reverse the process.

Formal View

fโˆ’1โ€‰โฃ:Yโ†’Xf^{-1}\colon Y \to X satisfies fโˆ’1(f(x))=xโ€…โ€Šโˆ€xโˆˆXf^{-1}(f(x)) = x\;\forall x \in X and f(fโˆ’1(y))=yโ€…โ€Šโˆ€yโˆˆYf(f^{-1}(y)) = y\;\forall y \in Y

Worked Examples

Example 1

easy
Find the inverse of f(x)=3xโˆ’7f(x) = 3x - 7.

Answer

fโˆ’1(x)=x+73f^{-1}(x) = \frac{x + 7}{3}

First step

1
Write y=3xโˆ’7y = 3x - 7.

Full solution

  1. 2
    Swap xx and yy: x=3yโˆ’7x = 3y - 7.
  2. 3
    Solve for yy: 3y=x+73y = x + 7, so y=x+73y = \frac{x + 7}{3}.
  3. 4
    Therefore fโˆ’1(x)=x+73f^{-1}(x) = \frac{x + 7}{3}.
To find an inverse, swap input and output then solve for the new output. The inverse 'undoes' the original function: applying ff then fโˆ’1f^{-1} returns the original input.

Example 2

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

Example 3

medium
Find the inverse of f(x)=3xโˆ’4x+2f(x) = \frac{3x - 4}{x + 2} and state its domain.

Common Mistakes

  • Reading fโˆ’1f^{-1} as a reciprocal - it means the reversing function, not 1/f1/f.
  • Finding an inverse without checking one-to-one - if ff fails the horizontal line test, no inverse exists.
  • Swapping xx and yy but forgetting to solve for the new yy - the inverse must be expressed as output in terms of input.

Why This Formula Matters

Inverses are how you solve f(x)=kf(x)=k exactly (logs invert exponentials, roots invert powers) and how you convert between paired quantities like Celsius and Fahrenheit. Without the one-to-one check, an 'inverse' would have to send one input to two outputs and could not be a function. Recognizing it by "If I know the output, does this rule hand back the exact input that produced it?" โ€” rather than by familiar numbers โ€” is what lets a student tell it apart from reciprocal and one-to-one mapping and function composition in a mixed problem set.

Frequently Asked Questions

What is the Inverse Function formula?

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.

How do you use the Inverse Function formula?

If ff turns aa into bb, then fโˆ’1f^{-1} turns bb back into aa. Reverse the process.

What do the symbols mean in the Inverse Function formula?

fโˆ’1f^{-1} denotes the inverse function. To find it: write y=f(x)y = f(x), swap xx and yy, solve for yy.

Why is the Inverse Function formula important in Math?

Inverses are how you solve f(x)=kf(x)=k exactly (logs invert exponentials, roots invert powers) and how you convert between paired quantities like Celsius and Fahrenheit. Without the one-to-one check, an 'inverse' would have to send one input to two outputs and could not be a function. Recognizing it by "If I know the output, does this rule hand back the exact input that produced it?" โ€” rather than by familiar numbers โ€” is what lets a student tell it apart from reciprocal and one-to-one mapping and function composition in a mixed problem set.

What do students get wrong about Inverse Function?

The procedure for inverse function is the easy part; the trap is reading fโˆ’1f^{-1} as a reciprocal. Asking "If I know the output, does this rule hand back the exact input that produced it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Inverse Function formula?

Before studying the Inverse Function formula, you should understand: function definition.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Functions and Graphs: Complete Foundations for Algebra and Calculus โ†’
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