Determine Where a Function is Increasing, Decreasing, or Constant

Analyzing the Toolkit Functions for Increasing or Decreasing Intervals

We will now return to our toolkit functions and discuss their graphical behavior in Figure 10, Figure 11, and Figure 12.

Function	Increasing/Decreasing	Example
Constant Function $f(x)=c$	Neither increasing nor decreasing
Identity Function $f(x)=x$	Increasing
Quadratic Function $f(x)=x^{2}$	Increasing on $(0, \infty)$ Decreasing on $(-\infty, 0)$ Minimum at $x=0$

Figure 10

Function	Increasing/Decreasing	Example
Cubic Function $f(x)=x^{3}$	Increasing
Reciprocal $f(x)=\frac{1}{x}$	Decreasing $(-\infty, 0) \cup(0, \infty)$
Reciprocal Squared $f(x)=\frac{1}{x^{2}}$	Increasing on $(-\infty, 0)$ Decreasing on $(0, \infty)$

Figure 11

Function	Increasing/Decreasing	Example
Cube Root $f(x)=\sqrt[3]{x}$	Increasing
Square Root $f(x)=\sqrt{x}$	Increasing on $(0, \infty)$
Absolute Value $f(x)=\|x\|$	Increasing on $(0, \infty)$ Decreasing on $(-\infty, 0)$

Figure 12