Intermediate Value Theorem

Definition: Intermediate Value Theorem

If $f$ is a continuous function on a closed interval $[a,b]$, and if ${\mathit{y}}_{\mathit{0}}$ is any value between $\mathit{f}\mathit{\left(}\mathit{a}\mathit{\right)}\mathit{ }\mathit{and }\mathit{f}\mathit{\left(}\mathit{b}\mathit{\right)}$, then ${\mathit{y}}_{\mathit{0}}\mathit{=}\mathit{f}\mathit{\left(}\mathit{c}\mathit{\right)}$ for some c in $[a,b]$.

Note:

–          This is a lot of fancy math talk to let you know that as long as your graph does not have any breaks in it (continuous), then you are guaranteed to get every y-value between the two end points or your equation. The intermediate value theorem is a natural extension of knowing that an equation is continuous (has no breaks).

–          Here is the same idea with some numbers attached.

–          If you start at the left endpoint y = 8 and draw to the right endpoint y = 18 , and you are not allowed to lift your pencil (continuous), you are guaranteed to draw through every y-value in between them. So, for example, you are guaranteed to get y = 14 .

–          And every one of those y-values will correspond to an x-value that is in your x-interval $[4,9]$. So, y = 14 came from x = 7 , which is guaranteed to be between the left end point x = 4 and right end point x = 9 .