# Intermediate value theorem review

Review the intermediate value theorem and use it to solve problems.

## What is the intermediate value theorem?

The intermediate value theorem describes a key property of continuous functions: for any function $f$ that's continuous over the interval $[a,b]$, the function will take any value between $f(a)$ and $f(b)$ over the interval.

More formally, it means that for any value $L$ between $f(a)$ and $f(b)$, there's a value $c$ in $[a,b]$ for which $f(c)=L$.

This theorem makes a lot of sense when considering the fact that the graphs of continuous functions are drawn without lifting the pencil. If we know the graph passes through $(a,f(a))$ and $(b,f(b))$...

... then it must pass through any $y$-value between $f(a)$ and $f(b)$.

*Want to learn more about the intermediate value theorem? Check out this video.*

## What problems can I solve with the intermediate value theorem?

Consider the continuous function $f$ with the following table of values. Let's find out where must there be a solution to the equation $f(x)=2$.

$x$ | $-2$ | $-1$ | $0$ | $1$ |
---|---|---|---|---|

$f(x)$ | $4$ | $3$ | $-1$ | $1$ |

Note that $f(-1)=3$ and $f(0)=-1$. The function must take any value between $-1$ and $3$ over the interval $[-1,0]$.

$2$ is between $-1$ and $3$, so there must be a value $c$ in $[-1,0]$ for which $f(c)=2$.

*Want to try more problems like this? Check out this exercise.*