Inverse concept, finding inverses, properties
High School Essentials • 9-12
If a function f takes an input and produces an output, the inverse function f-1 reverses the process: it takes the output and recovers the original input. Inverse functions appear throughout mathematics -- from solving equations to converting between units -- and understanding them deepens your grasp of what functions truly do.
Function f takes x to y: f(x) = y. The inverse function f-1 takes y back to x: f-1(y) = x. In other words:
These two equations are the defining properties of inverse functions. They "undo" each other completely.
Think of real-world examples: if f converts Celsius to Fahrenheit using f(C) = (9/5)C + 32, then f-1 converts Fahrenheit back to Celsius.
f-1(x) does not mean 1/f(x). The superscript -1 here denotes the inverse function, not a reciprocal. This is a common source of confusion. If you want the reciprocal of f(x), write [f(x)]-1 or 1/f(x).
Not every function has an inverse. A function has an inverse if and only if it is one-to-one (injective): each output value comes from exactly one input value. If two different inputs produce the same output, the function cannot be reversed (which input would the inverse return?).
The Horizontal Line Test provides a visual check: if every horizontal line intersects the graph at most once, the function is one-to-one and has an inverse.
One-to-one (has inverse):
NOT one-to-one (no inverse):
To find the inverse of a one-to-one function:
Find the inverse of f(x) = 4x - 7.
Verify: f(f-1(x)) = 4 * [(x + 7)/4] - 7 = (x + 7) - 7 = x. Confirmed.
Find the inverse of f(x) = (2x + 1) / (x - 3), where x is not 3.
f-1(x) = (3x + 1) / (x - 2), where x is not 2.
The graph of f-1 is the reflection of the graph of f over the line y = x. This makes geometric sense: swapping x and y in the equation is equivalent to reflecting across y = x.
To visualize: if (a, b) is on the graph of f, then (b, a) is on the graph of f-1. For example, if f(2) = 5, then f-1(5) = 2.
The function f passes through the points (1, 4), (2, 7), and (3, 10). Find three points on f-1.
The function f(x) = x2 is not one-to-one on all real numbers because f(3) = f(-3) = 9. It does not have an inverse unless you restrict the domain. For example, f(x) = x2 for x ≥ 0 is one-to-one, and its inverse is f-1(x) = sqrt(x). Always check that the function passes the horizontal line test before finding an inverse.
1. Find the inverse of f(x) = 5x + 2.
y = 5x + 2. Swap: x = 5y + 2. Solve: y = (x - 2)/5. So f-1(x) = (x - 2)/5.
2. Verify that f(x) = 3x - 9 and g(x) = (x + 9)/3 are inverses.
f(g(x)) = 3 * [(x + 9)/3] - 9 = (x + 9) - 9 = x.
g(f(x)) = (3x - 9 + 9)/3 = 3x/3 = x.
Since both compositions equal x, they are inverses.
3. Does f(x) = x2 - 4 (all real numbers) have an inverse? Why or why not?
No. f is not one-to-one: f(2) = 0 and f(-2) = 0. Two different inputs give the same output. It fails the horizontal line test.
4. Find the inverse of f(x) = (x - 1)3 + 2.
y = (x - 1)3 + 2. Swap: x = (y - 1)3 + 2. Then x - 2 = (y - 1)3. Take the cube root: y - 1 = (x - 2)1/3. So f-1(x) = (x - 2)1/3 + 1.
5. If the point (6, -1) is on the graph of f, what point must be on the graph of f-1?
Swap the coordinates: (-1, 6) is on the graph of f-1.
An inverse function reverses the input-output mapping of the original function. Only one-to-one functions (those passing the horizontal line test) have inverses. To find an inverse algebraically, swap x and y and solve for y. Graphically, f and f-1 are reflections over the line y = x. The compositions f(f-1(x)) and f-1(f(x)) both equal x, serving as the verification test for inverses. Remember that f-1(x) is notation for the inverse function, not the reciprocal.