What Is an Inverse Function?
An inverse function reverses the action of the original function. If a function f takes an input x and produces an output y, then the inverse function fâ»Â¹ takes y as input and returns the original x.
fâ»Â¹(f(x)) = x and f(fâ»Â¹(x)) = x — applying a function and its inverse in either order gives you back the starting value.Think of it as an "undo" button. If f(x) = 2x + 3 adds 3 and doubles, then fâ»Â¹(x) = (x â 3)/2 subtracts 3 and halves. The operations are reversed, and the order is reversed.
The notation fâ»Â¹ means the inverse function — it does not mean 1/f(x). This is a common source of confusion. The superscript â1 in function notation is about reversal, not reciprocals.
The Horizontal Line Test
Examples
| Function | Result | Why |
|---|---|---|
f(x) = 2x + 1 (a line) | Passes | Every horizontal line hits the graph exactly once. It has an inverse. |
f(x) = x² | Fails | The line y = 4 hits the graph at both x = 2 and x = â2. No inverse unless we restrict the domain to x ⥠0 or x ⤠0. |
f(x) = x³ | Passes | Every horizontal line hits the cubic exactly once. It has an inverse. |
f(x) = sin(x) | Fails | y = 0.5 has infinitely many solutions. No inverse unless restricted to [âÏ/2, Ï/2], which gives us arcsin. |
The horizontal line test is the graphical version of the one-to-one definition. On an exam, use it as a quick visual check before attempting to find the inverse algebraically.
How to Find the Inverse Function â Step by Step
Finding the inverse function algebraically follows a simple three-step procedure.
- Replace f(x) with y
This makes the algebra cleaner.
- Swap x and y
This reflects the function across the line
y = x, which is exactly what an inverse does graphically. - Solve for y
Isolate y on one side. The result is
fâ»Â¹(x).
Find the inverse of f(x) = 3x â 7.
Step 1: y = 3x â 7.
Step 2: x = 3y â 7.
Step 3: x + 7 = 3y, so y = (x + 7)/3.
fâ»Â¹(x) = (x + 7)/3Find the inverse of f(x) = â(x â 4) with domain x ⥠4.
Step 1: y = â(x â 4).
Step 2: x = â(y â 4).
Step 3: x² = y â 4, so y = x² + 4.
But we need the domain restriction: since the original range was y ⥠0, the inverse's domain is x ⥠0.
fâ»Â¹(x) = x² + 4 for x ⥠0Find the inverse of f(x) = (2x + 1)/(x â 3).
Step 1: y = (2x + 1)/(x â 3).
Step 2: x = (2y + 1)/(y â 3).
Step 3: Multiply both sides by (y â 3): x(y â 3) = 2y + 1. Expand: xy â 3x = 2y + 1. Collect y terms: xy â 2y = 3x + 1. Factor: y(x â 2) = 3x + 1. Solve: y = (3x + 1)/(x â 2).
fâ»Â¹(x) = (3x + 1)/(x â 2)How to Verify an Inverse
x. The two conditions are: f(fâ»Â¹(x)) = x and fâ»Â¹(f(x)) = x. If both hold, your inverse is correct.f(x) = 3x â 7 and fâ»Â¹(x) = (x + 7)/3.
Check 1: f(fâ»Â¹(x)) = f((x + 7)/3) = 3 · (x + 7)/3 â 7 = (x + 7) â 7 = x.
Check 2: fâ»Â¹(f(x)) = fâ»Â¹(3x â 7) = ((3x â 7) + 7)/3 = 3x/3 = x.
Both compositions return x, confirming the inverse is correct.This verification step is especially important for rational and radical functions where algebraic mistakes are common. It also makes for easy exam points — many teachers give partial credit for verification even if the inverse has a small error.
Domain and Range of Inverse Functions
f becomes the range of fâ»Â¹, and the range of f becomes the domain of fâ»Â¹. This swap is a direct consequence of swapping x and y.Key relationships
| Inverse property | Equals |
|---|---|
Domain of fâ»Â¹ | Range of f |
Range of fâ»Â¹ | Domain of f |
f(x) = x² with restricted domain [0, â).
Domain of f: [0, â).
Range of f: [0, â).
So fâ»Â¹(x) = âx has domain [0, â) and range [0, â).
Why restrictions matter
Without the domain restriction x ⥠0, the function x² is not one-to-one and has no inverse. The restriction forces the function to pass the horizontal line test.
This is exactly how inverse trig functions work: sin(x) is restricted to [âÏ/2, Ï/2] to create arcsin, cos(x) is restricted to [0, Ï] to create arccos, and tan(x) is restricted to (âÏ/2, Ï/2) to create arctan. Every inverse trig function exists because of a carefully chosen domain restriction on the original.
Graphing Inverse Functions
fâ»Â¹ is the reflection of the graph of f across the line y = x. This follows directly from the swap step: every point (a, b) on the graph of f becomes (b, a) on the graph of fâ»Â¹.How to graph an inverse
- Plot the original function
Start by drawing the graph of
f. - Draw the line y = x as a dashed reference line
This is the axis of reflection.
- Reflect each point across this line by swapping coordinates
Every point
(a, b)becomes(b, a).
Key observations
| Property of f | Result for fâ»Â¹ |
|---|---|
If the original function crosses the line y = x | The inverse also passes through that same point (the point is its own reflection). |
If f is increasing | fâ»Â¹ is also increasing. |
If f is decreasing | fâ»Â¹ is also decreasing. |
The graphs of f and fâ»Â¹ | Are always symmetric about y = x. |
For a quick visual check on an exam, plot two or three points of your inverse and verify they're reflections of the original. If they are, your algebra is likely correct. For more complex inverse function problems, scan them with Solver AI to get the full step-by-step solution with verification.