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