# 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 $f(b)$.
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 $f(c)=2$.