Inverse functions

What an inverse function does

An inverse function reverses another function. If a function takes an input $x$ and produces an output $y$, the inverse takes that output $y$ and returns the original input $x$.

If

$f(x) = y$

then the inverse function satisfies

$f^{-1}(y) = x$

The notation $f^{-1}$ means inverse function, not reciprocal. This is an important distinction. In general, $f^{-1}(x)$ does not mean $1/f(x)$.

A simple example

Consider

$f(x) = 3x + 2$

This function multiplies by 3 and then adds 2. To find the inverse, reverse those steps. Start with

$y = 3x + 2$

Subtract 2:

$y - 2 = 3x$

Divide by 3:

x = rac{y - 2}{3}

Now switch variable names if desired:

f^{-1}(x) = rac{x - 2}{3}

The inverse subtracts 2 and then divides by 3.

Inverses in physics

In physics, inverse functions appear whenever you want to solve a relationship in the opposite direction. If position is given as a function of time, $x(t)$, you might want to know the time when the object reaches a certain position. That requires an inverse relationship, at least over a suitable interval.

For example, if

$x(t) = vt$

with constant positive velocity, then

t(x) = rac{x}{v}

The original function answers: where is the object at time $t$? The inverse answers: when does the object reach position $x$?

One-to-one requirement

Not every function has an inverse over its entire domain. A function must be one-to-one, meaning each output comes from only one input. If two different inputs produce the same output, the inverse would not know which input to return.

For example,

$f(x) = x^2$

is not one-to-one over all real numbers because $f(2)=4$ and $f(-2)=4$. The output 4 corresponds to two inputs. To define an inverse, we often restrict the domain. If we use only $x geq 0$, then the inverse is

$f^{-1}(x) = sqrt{x}$

This idea matters in physics because physical context often selects the correct branch. Speed may be nonnegative, time may be positive, and distance may be restricted.

Graphing inverses

The graph of an inverse function is the reflection of the original graph across the line

$y = x$

This happens because inputs and outputs switch roles. A point $(a,b)$ on the original function corresponds to $(b,a)$ on the inverse.

This graphical view helps you understand why a function that fails the horizontal line test does not have a full inverse. If a horizontal line crosses the original graph more than once, then one output corresponds to multiple inputs.

Inverse trig and inverse exponentials

Many important inverse functions appear in physics. Logarithms are inverses of exponentials. Inverse trigonometric functions, such as $sin^{-1}(x)$, help find angles from ratios. Square roots are inverses of squaring when the domain is restricted.

For example, if

sin( heta) = rac{opposite}{hypotenuse}

then

ight)$$ ## The big idea Inverse functions reverse relationships. In physics, they let you switch from predicting an output to finding the input that caused it. Understanding inverses helps with solving equations, interpreting graphs, finding times from positions, finding angles from trig ratios, and solving exponential models with logarithms.