What Is an Inverse Function?
An inverse function undoes the original: if f(a) = b then f⁻¹(b) = a. Definition, notation, the graph reflection across y = x, and real-world examples.
What Is an Inverse Function? The Definition You Need
An inverse function undoes another function, so what is an inverse function? If f(a) = b, then the inverse f⁻¹(b) = a. The notation f⁻¹(x) does not mean 1/f(x); that is the reciprocal, not the inverse. This confusion is the most common mistake in Algebra 2 and Precalculus. OpenStax Precalculus 2e covers this in section 1.7, "Inverse Functions."
Start with a simple example: f(x) = 2x + 3. Input x = 1, f(1) = 5. The inverse f⁻¹(5) should return 1. f⁻¹(x) = (x - 3)/2 works: f⁻¹(5) = (5-3)/2 = 1. Composition confirms it: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for all x in the domain.
Inverse Function Notation: f⁻¹(x) Is Not the Reciprocal
f⁻¹(x) versus 1/f(x). The superscript -1 in f⁻¹ is inverse notation, not an exponent. It means "the function that reverses f." The expression 1/f(x) is the multiplicative reciprocal. These are not the same. For f(x) = 2x, f⁻¹(x) = x/2, while 1/f(x) = 1/(2x). Only one of them undoes f.
Why this matters. A student who interprets f⁻¹(x) as the reciprocal will get every inverse problem wrong. On tests, check the notation before you start. OpenStax Precalculus 2e section 1.7 uses f⁻¹ exclusively for the inverse function.
Input-Output Tables Show the Inverse Swap
An input-output table makes the inverse relationship obvious. If f maps 1→5, 2→7, 3→9, then the inverse table swaps the rows: 5→1, 7→2, 9→3. Every output of f becomes an input of f⁻¹.
How to build one. Write down a few (x, f(x)) pairs for your function. Then write a new table where the first column is f(x) and the second column is x. That second table is the inverse. This method works whether you have the formula or just a set of points, useful on tests when you cannot find the formula.
| x (input) | f(x) = 2x + 3 | f⁻¹(x) = (x – 3)/2 |
|---|---|---|
| 0 | 3 | — |
| 1 | 5 | — |
| — | 3 | 0 |
| — | 5 | 1 |
Graphs Reflect Across y = x
The graph of an inverse function is the mirror image of the original function across the line y = x. Fold the graph along that diagonal; the two curves match exactly. This follows from swapping the coordinates.
Checking your work visually. Plot the original function, then plot f⁻¹. They should be symmetric about y = x. If they are not, the inverse is wrong. The function f(x) = 1/x is its own inverse because it is symmetric about y = x without any folding needed.
What a non-inverse looks like. For f(x) = x², the graph after swapping x and y gives two y-values for one x (the sideways parabola). It fails the vertical line test, so it is not a function. That is why x² needs a domain restriction to have an inverse.
Which Functions Have Inverse Functions
Functions That Pass the Horizontal Line Test
Only one-to-one functions have inverses. A function is one-to-one if each output comes from exactly one input. The horizontal line test tells you: if any horizontal line crosses the graph more than once, the function is not one-to-one and has no inverse function (without restriction).
Functions that pass (no restriction needed):
- Linear functions f(x) = ax + b (a ≠ 0), always one-to-one.
- Exponential functions f(x) = a·bˣ, one-to-one on all reals.
- Logarithmic functions f(x) = log_b(x), one-to-one on (0, ∞).
- Odd-degree polynomials like f(x) = x³, one-to-one on all reals.
- Rational functions f(x) = (ax+b)/(cx+d), one-to-one on each branch.
Functions That Need a Domain Restriction
- Quadratic functions f(x) = x², not one-to-one on (−∞, ∞). Restrict to [0, ∞) or (−∞, 0] to get an inverse.
- Trigonometric functions sin(x), cos(x), tan(x), periodic, not one-to-one. Restrict to principal value ranges: arcsin uses [−π/2, π/2], arccos uses [0, π], arctan uses (−π/2, π/2). OpenStax Precalculus 2e section 1.8 details these restrictions.
- Absolute value f(x) = |x|, not one-to-one. No inverse without splitting the domain.
Real-World Inverse Function Examples
Temperature Conversion
Convert Fahrenheit to Celsius: C = (F - 32) × 5/9. The inverse converts Celsius back to Fahrenheit: F = C × 9/5 + 32. Every temperature has exactly one Fahrenheit value and one Celsius value, so the function is one-to-one.
Logarithms and Exponentials
Exponential growth f(x) = 2ˣ and its inverse log₂(x) are used in measuring pH, sound intensity in decibels, and earthquake magnitudes. If you know the pH (a logarithmic scale), the inverse exponential gives the hydrogen ion concentration.
Inverse Trigonometric Functions
arcsin(x), arccos(x), arctan(x) find angles from side lengths. If sin(θ) = 0.5, then θ = arcsin(0.5) = π/6 (30°). The principal value ranges are fixed by convention: arcsin outputs in [−π/2, π/2], arccos in [0, π], arctan in (−π/2, π/2).
Common Questions
What is an inverse function in simple terms?
It undoes the original function. If f turns 2 into 5, the inverse f⁻¹ turns 5 back into 2. The notation f⁻¹ means inverse, not reciprocal.
How do you calculate the inverse of a function?
Write y = f(x), swap x and y, then solve for y. Rename y as f⁻¹(x). Verify by checking f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
Why do some functions not have an inverse?
They fail the horizontal line test. A function must be one-to-one (each output from one input) to have an inverse. Quadratic functions need a domain restriction to become one-to-one.
How do I verify my inverse is correct?
Use composition. If f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for all x in the domains, the inverse is correct. This is the only reliable check.