Examples · Graphing
Inverse Function Examples: 9 Worked Problems
Nine inverses, ordered so each one adds exactly one new idea to the last — and every result here is checked by composing it back with the original function.
Linear functions, the simplest case
The first two examples need nothing but undoing addition, subtraction, multiplication and division, in reverse order.
- y = 3x − 4write it as y = f(x)
- (y + 4) / 3 = xadd 4, then divide by 3 — undoing in reverse order
- f⁻¹(x) = (x + 4) / 3swap x and y
- y = (x + 2) / 5write it as y = f(x)
- 5y − 2 = xmultiply by 5, then subtract 2
- f⁻¹(x) = 5x − 2swap x and y
Notice the two examples undo each other's structure: the first ends in division, the second starts by undoing one. Checking is one substitution: f(f⁻¹(2)) should return 2, and it does for both.
A square root, undone by squaring
- y = √(x − 3)write it as y = f(x)
- y² = x − 3undo the square root: square both sides
- y² + 3 = xadd 3
- f⁻¹(x) = x² + 3swap x and y
A cube, undone by a cube root
- y = x³ + 1write it as y = f(x)
- y − 1 = x³subtract 1
- (y − 1)^(1/3) = xundo the cube: take the cube root of both sides
- f⁻¹(x) = (x − 1)^(1/3)swap x and y
Unlike the square root case, no domain restriction is needed here — every real number has exactly one real cube root, so x³ is one-to-one over all real numbers on its own.
Exponentials, undone by a logarithm
- y = e^(2x)write it as y = f(x)
- ln y = 2xundo the exponential: take ln of both sides
- ln(y) / 2 = xdivide by 2
- f⁻¹(x) = ln(x) / 2swap x and y
- y = 2^xwrite it as y = f(x)
- log₂ y = xundo the exponential: take log base 2 of both sides
- f⁻¹(x) = log₂ x = ln(x) / ln(2)swap x and y; log base 2 written using natural log
Any base follows the same pattern: aˣ is undone by log base a, and log base a can always be rewritten as ln(x) / ln(a) when a calculator only has a natural log key.
A logarithm, undone by an exponential
- y = ln(x + 1)write it as y = f(x)
- e^y = x + 1undo the logarithm: exponentiate both sides
- e^y − 1 = xsubtract 1
- f⁻¹(x) = e^x − 1swap x and y
This is example 5 in reverse, in a sense — ln and eˣ undo each other in both directions, which is exactly why composing them, in either order, returns x.
A higher power, undone by a fractional exponent
- y = x⁵write it as y = f(x)
- y^(1/5) = xundo the fifth power: take the fifth root of both sides
- f⁻¹(x) = x^(1/5)swap x and y
Odd powers behave like cubes — one-to-one over all real numbers, no domain restriction needed. An even power here would need the same x ≥ 0 restriction seen in example 3.
A reciprocal shape, undone by rearranging
- y = 1 / (x + 1)write it as y = f(x)
- 1 / y = x + 1undo the division: take the reciprocal of both sides
- 1 / y − 1 = xsubtract 1
- f⁻¹(x) = 1 / x − 1swap x and y
Checking every result the same way
Compose the candidate inverse with the original and confirm the input comes back — that composition of functions returning x is the definition of an inverse, not just a nice extra step.
| Example | f(x) | f⁻¹(x) | Composition check |
|---|---|---|---|
| 1 | 3x − 4 | (x + 4) / 3 | f(f⁻¹(2)) = 2 ✓ |
| 5 | e^(2x) | ln(x) / 2 | f(f⁻¹(4)) = 4 ✓ |
| 8 | x⁵ | x^(1/5) | f(f⁻¹(3)) = 3 ✓ |
This is exactly the check the Inverse Function Calculator on this site runs numerically before showing any answer — composing the candidate inverse with the original and comparing the result to the input, not just trusting the algebra.
What all nine worked examples have in common
Every one of these worked examples follows the same shape: write y = f(x), undo the operations from the outside in, solve for x, then swap x and y. The variety above — linear, root, cube, exponential, log, power, reciprocal — is really the same three-step method applied to nine different chains of operations, and once the pattern is familiar, a tenth kind of function is rarely harder than these nine, only unfamiliar.
Examples 3 and 8 are the two to remember for domain: a restricted domain shows up wherever the original function is not naturally one-to-one over all real numbers, and the restriction always comes from the original function's own range, not from anything invented for the inverse.
Questions about inverse function examples
What is the inverse of f(x) = 3x − 4?
(x + 4) / 3. Add 4 to undo the subtraction, then divide by 3 to undo the multiplication — undone in the reverse of the order they were applied.
What is the inverse of an exponential like 2^x?
A logarithm base 2: log₂(x), which can be written ln(x) / ln(2) using natural log. Exponentials and logarithms undo each other by definition.
Why does x³ not need a domain restriction, but x² does?
Because every real number has exactly one real cube root, so x³ is one-to-one over all real numbers already. x² is not — both 2 and −2 square to 4 — so its inverse √x only applies once the domain is restricted to x ≥ 0.
How do I check an inverse function is correct?
Compose it with the original: f(f⁻¹(x)) should simplify back to x. If it does not, an operation was undone in the wrong order.