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Inverse Functions Calculator

Find the inverse of a linear function f(x) = mx + b. Enter slope m and y-intercept b to get the inverse function f⁻¹(x) with the swap-and-solve method shown.

f(x) = m·x + b

Result

f⁻¹(x) = 0.5x - 1.5

Quick Answer

An inverse functions calculator finds f⁻¹(x) for a linear function f(x) = mx + b by swapping x and y, then solving for the new y. The result is f⁻¹(x) = (1/m)x − (b/m). For f(x) = 3x + 6, f⁻¹(x) = (1/3)x − 2. Check: f(f⁻¹(x)) = 3((1/3)x − 2) + 6 = x − 6 + 6 = x. Confirmed.

Linear Function Inverses: What This Calculator Finds

This calculator finds the inverse of a linear function of the form f(x) = mx + b, where m is the slope and b is the y-intercept. It is scoped to linear (degree-1) functions only. The general method for finding any inverse (swap x and y, then solve for y) is described conceptually below, but the calculator implements and displays working for the linear case only.

The Inverse Function Formula for f(x) = mx + b

f⁻¹(x) = (1/m)x − (b/m) Derivation: y = mx + b → x = my + b → y = (x − b)/m

  • Swap x and y: x = my + b
  • Solve for y: y = (x − b) / m
  • Inverse function: f⁻¹(x) = (1/m)x − (b/m)
  • Identity check: f(f⁻¹(x)) = x

How To Find the Inverse of a Linear Function

Inputs

  • m: the slope (coefficient of x) in f(x) = mx + b
  • b: the y-intercept (constant term) in f(x) = mx + b

Steps

  1. Write f(x) = mx + b and identify m and b.
  2. Enter m and b into the calculator.
  3. Read f⁻¹(x) as (1/m)x − (b/m).
  4. Verify the inverse by substituting a specific x-value.

Inverse Function Worked Example: f(x) = −4x + 12

Finding the inverse of f(x) = −4x + 12.

  1. Write forward function: y = −4x + 12.
  2. Swap x and y: x = −4y + 12.
  3. Solve for y: x − 12 = −4y → y = −(1/4)x + 3.
  4. Check at x = 4: f(4) = −4, f⁻¹(−4) = 4.

f⁻¹(x) = −(1/4)x + 3.

When Inverse Functions Exist and When They Do Not

A function has an inverse only if it is one-to-one. Linear functions with m ≠ 0 are always one-to-one and always have an inverse. If m = 0, f(x) = b is a constant function with no inverse.

Assumptions

  • f(x) = mx + b with m ≠ 0.
  • The domain and range are all real numbers.
  • The inverse formula is the algebraic inverse.

Limitations

  • Linear (degree-1) functions only; does not find inverses of quadratic, exponential, logarithmic, or trigonometric functions.
  • m = 0 produces no inverse.
  • Does not graph f(x) and f⁻¹(x).

In Practice

The most common application of the linear inverse is converting between two linear measurement scales. Temperature conversion (Celsius to Fahrenheit is f(x) = (9/5)x + 32; the inverse is f⁻¹(x) = (5/9)(x − 32)) is a classic example. Any time you have a linear formula and need to run it backwards, starting from the output and recovering the input, you need the inverse function.

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Frequently Asked Questions: Inverse Functions Calculator

What is an inverse function?

An inverse function f⁻¹ reverses the operation of f: if f(a) = b, then f⁻¹(b) = a. The composition f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for all valid x. An inverse exists when f is one-to-one.

How does the swap-and-solve method work?

To find the inverse: (1) write y = f(x), (2) swap x and y to get x = f(y), (3) solve the resulting equation for y. The y you solve for is f⁻¹(x).

What does the graph of an inverse function look like?

The graph of f⁻¹ is the reflection of the graph of f across the line y = x. If a point (a, b) is on f, then (b, a) is on f⁻¹.

Can every function be inverted?

No. Only one-to-one (injective) functions have inverses over their entire domain. Constant functions (m = 0) and even-degree polynomials over their full domain cannot be inverted without domain restrictions.

What is the difference between f⁻¹(x) and 1/f(x)?

These are completely different. f⁻¹(x) is the inverse function that undoes f. 1/f(x) is the reciprocal of f(x): the number 1 divided by the output of f. They are not related.

Sources

Last updated: . Reviewed for accuracy against the formula shown above.