Chapter 3.5, Problem 33E

a.

To determine

To describe: the behavior of even $f\left(x\right)$ for $x<0$ and sketch its graph.

Answer to Problem 33E

  $f\left(x\right)\to -\infty \text{ as }x\to \infty$

Explanation of Solution

Given information:

Function $f\left(x\right)$ increases when $0<x<2$ , and decreases at $x>2$ .

Jump discontinuity at $x=2$ .

  $f\left(x\right)\to -\infty$ as $x\to \infty$

Calculation:

An even function has symmetry along the y axis. So the end behavior to the right and left will be the same, i.e. as $x$ approaches to $\infty$ the $f\left(x\right)$ approaches to $-\infty$ . The graph will be as follows:

  

b.

To determine

To describe: the behavior of odd $f\left(x\right)$ for $x<0$ and sketch its graph.

Answer to Problem 33E

  $f\left(x\right)\to \infty \text{ as }x\to -\infty$

Explanation of Solution

Given information:

Function $f\left(x\right)$ increases when $0<x<2$ , and decreases at $x>2$ .

Jump discontinuity at $x=2$ .

  $f\left(x\right)\to -\infty$ as $x\to \infty$

Calculation:

An odd function has $f\left(-x\right)=-f\left(x\right)$ . So the end behavior to the right and left will be the totally opposite to each other. Thus the end behavior of given odd function is that as $x$ approaches to $-\infty$ the $f\left(x\right)$ approaches to $\infty$ . The graph will be as follows: