Determine Where a Function is Increasing, Decreasing, or Constant

Now that we have more practice graphing and working with equations of functions, we will learn how to describe the behavior of a function over a large interval or by zooming in on a local area where the function's behavior changes.

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