Inverse Functions

You’ll be able to

Explain how exponential and logarithmic functions are inverses
Find the inverse of a basic exponential function
Use the fact that composing a function with its inverse returns the input

Inverses undo each other

Two functions are inverses when each one undoes the other: applying f and then f⁻¹ returns you to your starting input. Graphically, inverse functions are reflections of each other across the line y = x — the roles of input and output are swapped. To find an inverse algebraically, swap x and y in the equation and solve for y.

Exponentials and logarithms are inverses

The exponential f(x) = bˣ and the logarithm g(x) = log_b(x) are a perfect inverse pair. The exponential takes an exponent to a value; the logarithm takes the value back to the exponent. So the inverse of f(x) = 2ˣ is f⁻¹(x) = log₂(x). This is why a log “cancels” an exponential of the same base, and vice versa.

Inverse composition identity

f(f⁻¹(x)) = x and f⁻¹(f(x)) = x

Composing a function with its inverse (in either order, on the appropriate domain) returns the original input. For example, 2^(log₂ x) = x.

Worked example

Find the inverse of f(x) = 2ˣ.

1.Write the function as y = 2ˣ.
2.Swap x and y to reflect across y = x: x = 2ʸ.
3.Solve for y by taking the base-2 logarithm of both sides: log₂(x) = y.
4.So the inverse is f⁻¹(x) = log₂(x).

Answer: f⁻¹(x) = log₂(x). Swapping x and y and solving turns the exponential into its inverse logarithm of the same base.

Checkpoint

What is the inverse of f(x) = 2ˣ?

Watch out

Do not confuse the inverse f⁻¹ with the reciprocal 1/f. The inverse of 2ˣ is log₂(x), not 2^(−x) or (1/2)ˣ. Reflecting across y = x is a different operation from flipping the sign of the exponent.

Checkpoint

If f and f⁻¹ are inverse functions, what is f(f⁻¹(x))?

On the exam

Use the composition identity to simplify: e^(ln x) = x and log(10ˣ) = x. Spotting a function next to its inverse lets you collapse a messy expression in one step.

Answer the 2 checkpoints as you read.