Inverse Function Formula

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

f−1(f(x))=x and f(f−1(x))=x

When to use: If f turns a into b, then f−1 turns b back into a. Reverse the process.

Quick Example

f(x)=2x (double).
f−1(x)=x2 (halve).
f−1(f(3))=f−1(6)=3.

Notation

f−1 denotes the inverse function. To find it: write y=f(x), swap x and y, solve for y.

What This Formula Means

The inverse of a function f is a function f−1 that reverses f: if f(a)=b then f−1(b)=a. It exists only when f is one-to-one.

If f turns a into b, then f−1 turns b back into a. Reverse the process.

Formal View

f−1 ⁣:Y→X satisfies f−1(f(x))=x  ∀x∈X and f(f−1(y))=y  ∀y∈Y

Worked Examples

Example 1

easy
Find the inverse of f(x)=3x−7.

Answer

f−1(x)=x+73

First step

1
Write y=3x−7.

Full solution

  1. 2
    Swap x and y: x=3y−7.
  2. 3
    Solve for y: 3y=x+7, so y=x+73.
  3. 4
    Therefore f−1(x)=x+73.
To find an inverse, swap input and output then solve for the new output. The inverse 'undoes' the original function: applying f then f−1 returns the original input.

Example 2

medium
Find the inverse of f(x)=2x+5x−1 for x≠1.

Example 3

medium
Find the inverse of f(x)=3x−4x+2 and state its domain.

Common Mistakes

  • Reading f−1 as a reciprocal - it means the reversing function, not 1/f.
  • Finding an inverse without checking one-to-one - if f fails the horizontal line test, no inverse exists.
  • Swapping x and y but forgetting to solve for the new y - the inverse must be expressed as output in terms of input.

Why This Formula Matters

Inverses are how you solve f(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?" — rather than by familiar numbers — 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 f is a function f−1 that reverses f: if f(a)=b then f−1(b)=a. It exists only when f is one-to-one.

How do you use the Inverse Function formula?

If f turns a into b, then f−1 turns b back into a. Reverse the process.

What do the symbols mean in the Inverse Function formula?

f−1 denotes the inverse function. To find it: write y=f(x), swap x and y, solve for y.

Why is the Inverse Function formula important in Math?

Inverses are how you solve f(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?" — rather than by familiar numbers — 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 f−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.

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