One-to-One Functions

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is called one-to-one if $f(x_{1})=f(x_{2})$ implies $x_{1}=x_{2}$, where $x_{1},x_{2}\in \mathbb{R}$. To show a function is one-to-one:

1. Begin with $f(x_{1})=f(x_{2})$.
2. Evaluate $f$ at $x_{1}$ and $f$ at $x_{2}$ and set these expressions equal to each other.
3. Use algebra to simplify to $x_{1}=x_{2}$.

Inverse Functions

If $f:D \rightarrow R$ is a one-to-one function with domain $D$ and range $R$, then the inverse function $f^{-1}:R \rightarrow D$ has domain $R$ and range $D$ and is defined by $$ f^{-1}(y) = x \Leftrightarrow f(x)=y.$$ for all $y \in R$.
Note: $\displaystyle f^{-1}(x) \neq \frac{1}{f(x)}$. We use $\displaystyle [f(x)]^{-1}=\frac{1}{f(x)}$ to denote the reciprocal of $f(x)$.
If $f(x)$ and $f^{-1}(x)$ are inverse functions, they satisfy the inverse function properties:

1. $(f \circ f^{-1})(x) = f(f^{-1}(x)) = x$ for all $x \in R$.
2. $(f^{-1} \circ f)(x) = f^{-1}(f(x)) = x$ for all $x \in D$.

Finding an Inverse Function

If $f(x)$ is a one-to-one function, we can find its inverse by using the following procedure.

1. Write $y=f(x)$.
2. Interchange $x$ and $y$.
3. Solve the equation for $y$.
4. The resulting equation is $y = f^{-1}(x)$.