What is an inverse function, and how do you find one?

Think of a function as a machine. f(x) = 2x + 1 takes 3, doubles it to 6, adds 1 and gives 7. The inverse machine runs the other way: it takes 7, subtracts 1 to get 6, halves it and gives back 3. Every step is undone, in the reverse order, like taking off your shoes before your socks.

The inverse of f is written f⁻¹(x) and read "f inverse of x". The little −1 is not a power: f⁻¹(x) does not mean 1 ÷ f(x).

To find an inverse with algebra, use the swap method:

1. Write y = f(x), for example y = 2x + 1.
2. Swap x and y: x = 2y + 1.
3. Solve for y: x − 1 = 2y, so y = (x − 1)/2.

So f⁻¹(x) = (x − 1)/2. Check: f(3) = 7 and f⁻¹(7) = 6/2 = 3. In general, f(f⁻¹(x)) = x and f⁻¹(f(x)) = x: putting a function and its inverse together gives back what you started with.

On a graph, swapping x and y turns every point (a, b) into (b, a). That is a reflection in the diagonal line y = x. If (3, 7) is on the graph of f, then (7, 3) is on the graph of f⁻¹.

Not every function has an inverse. f(x) = x² sends both 3 and −3 to 9, so an inverse would not know which one to give back. A function has an inverse only if it is one-to-one: each output comes from exactly one input. On a graph, use the horizontal line test: if a horizontal line crosses the graph more than once, there is no inverse. The fix is to restrict the domain. For x ≥ 0, f(x) = x² is one-to-one and its inverse is f⁻¹(x) = √x.

The inputs and outputs also swap: the domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.

You already know several famous pairs: x² and √x (for x ≥ 0), x³ and ∛x, 10ˣ and log₁₀ x (see logarithms), eˣ and ln x, and sin x and sin⁻¹ x (also called arcsin) on a restricted domain.

For a quadratic, restrict the domain first. f(x) = (x − 3)² + 1 with x ≥ 3: swap to x = (y − 3)² + 1, so (y − 3)² = x − 1 and y = 3 + √(x − 1). Take the positive root because y ≥ 3. So f⁻¹(x) = 3 + √(x − 1) for x ≥ 1. Check: f(4) = 2 and f⁻¹(2) = 3 + 1 = 4.

What is an inverse function in simple words?#

It is the function that reverses another one. If f turns 3 into 7, the inverse turns 7 back into 3. Doing f and then f⁻¹ gets you back where you started.

What does f⁻¹(x) mean?#

It is the notation for the inverse of f, read "f inverse of x". It is not a power and it does not mean 1/f(x).

How do you find the inverse of a function?#

Write y = f(x), swap x and y, and rearrange to make y the subject. The result is f⁻¹(x). Always check it with a number.

How do you graph an inverse function?#

Reflect the graph of f in the line y = x: each point (a, b) moves to (b, a). In Desmos, if you have defined f(x), typing x = f(y) draws the inverse.

Does every function have an inverse?#

No. Only one-to-one functions do, the ones that pass the horizontal line test. Others, like x², get an inverse once you restrict the domain.

How do you find the inverse of a quadratic?#

Restrict the domain so only one side of the parabola is used, then swap x and y and solve, often by completing the square. Keep only the square root that matches the restricted domain.

What are inverse trig functions?#

sin⁻¹, cos⁻¹ and tan⁻¹ (arcsin, arccos, arctan) give the angle from a ratio. Because sin repeats, its inverse uses a restricted domain: sin⁻¹(0.5) = 30°.

What is the derivative of an inverse function?#

In calculus, (f⁻¹)′(x) = 1 ÷ f′(f⁻¹(x)). For f(x) = x³, f⁻¹(8) = 2 and f′(2) = 12, so the slope of ∛x at x = 8 is 1/12.