11th Grade Q1

The inverse of a one-to-one function "reverses" the function. If $f(x)$ maps an input $x$ to an output $y$, then the inverse function, denoted $f^{-1}(x)$, maps $y$ back to $x$.

For a function to have an inverse, the function must be one-to-one (each output corresponds to only one input).

If $f(x)$ has domain $X$ and co-domain $Y$, for $f^{-1}(x)$ the domain is $Y$ and the co-domain is $X$.

How to Find the Inverse

To find the inverse:

1. Replace $f(x)$ with $y$:

1. Swap $x$ and $y$:

1. Solve for $y$ in terms of $x$. This new expression is $f^{-1}(x)$.

Example 1

Let's find the inverse of $f(x)=4x-2$.

1. Replace $f(x)$ with $y$

2. Swap $x$ and $y$:

3. Solve for $y$:

So, the inverse is

Example 2

Consider the function $f(x)=x^2-1$. Is it invertible? Remember, a function is invertible only if it is one-to-one.

Compute $f(1)$:

Compute $f(-1)$:

Here, $f(1)=f(-1)$, meaning different inputs produce the same output. Therefore, $f(x)=x^2-1$ is not one-to-one and cannot have an inverse.