Inverse Function Calculator

Find the inverse of a linear function f(x) = ax + b. Verify that f(f⁻¹(x)) = x. Understand when functions have inverses and how to swap x and y to find f⁻¹(x).

∑ Algebra📐 f⁻¹(x) = (x - b) / a🔢 Math

Coefficient a in f(x) = ax + b

Constant b

Verify: evaluate f⁻¹ at x =

Please enter valid values.

Formula & Reference

Variable	Symbol	Formula	Units
Inverse Function Calculator	—	f⁻¹(x) = (x - b) / a	function

Step-by-Step Examples

Example 1

Linear Inverse

f(x) = 3x + 5. Find f⁻¹(x).

Set y = 3x+5, swap: x = 3y+5
Solve for y: y = (x−5)/3
f⁻¹(x) = (x−5)/3
Check: f(f⁻¹(11)) = f(2) = 11 ✓

✓ f⁻¹(x) = (x−5)/3

Example 2

Verify Inverse

f(x)=2x−4. f⁻¹(x) = (x+4)/2. Check f(f⁻¹(6)).

f⁻¹(6) = (6+4)/2 = 5
f(5) = 2(5)−4 = 6
f(f⁻¹(6)) = 6 ✓

✓ Verified: f(f⁻¹(6)) = 6

Example 3

Negative Slope

f(x) = −2x + 8. Find inverse.

y = −2x+8, swap: x = −2y+8
y = (8−x)/2 = −(1/2)x + 4
f⁻¹(x) = −x/2 + 4

✓ f⁻¹(x) = −x/2 + 4

Real-World Applications

📚

Algebra

Finding inverses is a fundamental concept in precalculus and algebra 2.

💻

Cryptography

Encryption and decryption use inverse functions: decrypt(encrypt(x)) = x.

🧪

Physics

Converting between units uses inverse functions (°F to °C and back).

📊

Data Science

Logit and sigmoid are inverse functions used in logistic regression.

Common Mistakes to Avoid

⚠️

Not swapping x and y

To find inverse: replace f(x) with y, SWAP x and y, then solve for y. Forgetting the swap is the most common error.

⚠️

All functions have inverses — FALSE

Only one-to-one (injective) functions have inverses. f(x)=x² does not have an inverse over all reals (fails horizontal line test).

⚠️

The −1 notation

f⁻¹(x) means inverse function, NOT 1/f(x). These are completely different: f⁻¹(x) ≠ [f(x)]⁻¹

Frequently Asked Questions

What is an inverse function? ▾

A function that undoes another. If f(a)=b, then f⁻¹(b)=a. Applying f then f⁻¹ returns to the start: f⁻¹(f(x)) = x.

What is the horizontal line test? ▾

A function has an inverse only if every horizontal line crosses its graph at most once (one-to-one). f(x)=x² fails this test.

How to find an inverse algebraically? ▾

1) Replace f(x) with y. 2) Swap x and y. 3) Solve for y. 4) Replace y with f⁻¹(x).

What is the graph of an inverse? ▾

The graph of f⁻¹ is the reflection of the graph of f over the line y=x.

Do exponential and log functions have inverses? ▾

Yes, and they are inverses of each other: f(x)=eˣ and f⁻¹(x)=ln(x). Similarly, f(x)=10ˣ and f⁻¹(x)=log₁₀(x).

Related Math Calculators

Formula Explorer connections

Interpretation: This formula describes an algebraic relationship among variables, functions, equations, roots or sequences. Assumption: Respect the expression’s domain and excluded values. Check roots in the original equation because transformations can introduce extraneous solutions.

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