One-to-One Functions and Inverse Functions

One-to-one functions are super important when we talk about inverse functions. Here’s why:

1. What is a One-to-One Function?
   A function, which we can call $f(x)$, is one-to-one if each unique input gives a unique output. This means if you have two different numbers, $a$ and $b$, and they produce the same output, then those two numbers must be the same. In simpler terms, if $f(a) = f(b)$, then $a$ must equal $b$. This helps make sure that every output is tied to just one input.

2. Why Do We Need One-to-One Functions for Inverses?
   For a function to have an inverse, it needs to be one-to-one. If it’s not, then different inputs could give us the same output. This would make it hard, or even impossible, to define an inverse. For example, consider the function $f(x) = x^2$. If you plug in $-2$ and $2$, both give you the same output of $4$. So, you can’t find a single inverse for this function since it doesn’t satisfy the one-to-one condition.

3. How to Find Inverses
   When $f(x)$ is a one-to-one function, we can find its inverse, which we can call $f^{-1}(x)$. To do this, we switch $x$ and $y$ in the equation $y = f(x)$ and then solve for $x$. For instance, if we have $f(x) = 3x + 2$, we can find the inverse like this:

   • Start with $y = 3x + 2$.
   • Switch $x$ and $y$: $x = 3y + 2$.
   • Now, solve for $y$:
     • First, subtract $2$ from both sides: $3y = x - 2$.
     • Next, divide by $3$: $y = \frac{x - 2}{3}$.
   • So, the inverse function is $f^{-1}(x) = \frac{x - 2}{3}$.

4. Seeing it on a Graph
   You can tell if a function is one-to-one by using something called the Horizontal Line Test. If you draw any horizontal line on the graph, it should only hit the curve at one point. If it touches more than once, then the function isn’t one-to-one, and it won’t have an inverse.

In short, one-to-one functions are really important for figuring out and finding inverse functions in algebra.