Q 92. - Precalculus Enhanced with Graphing Utilities

Question

Suppose that a friend of yours does not understand the idea of increasing and decreasing functions. Provide an explanation, complete with graphs, that clarifies the idea.

Step-by-Step Solution

Answer

Increasing function: 

A function $f$ is said to be increasing on an interval $I$ if there exists $x$ and $y$ such that $x<y$, then $f\left(x\right)\le f\left(y\right)$.

Decreasing function: 

A function $f$ is said to be decreasing on an interval $I$ if there exists $x$ and $y$ such that $x<y$, then $f\left(x\right)\ge f\left(y\right)$.

Step 1. Define increasing function along with graph.

A function $f\left(x\right)$ is said to be increasing on an interval $I$ if there exists two numbers $x$ and $y$ in $I$ such that $x<y$, then $f\left(x\right)\le f\left(y\right)$.

The required graph is shown below:

Step 2. Define decreasing function along with graph.

A function $f\left(x\right)$ is said to be decreasing on an interval $I$ if there exists two numbers $x$ and $y$ in $I$ such that $x<y$, then $f\left(x\right)\ge f\left(y\right)$.

The required graph is shown below: