Chapter 1.3, Problem 33E

(a)

To determine

To sketch: the graph of the given function.

Explanation of Solution

Given information:

The function $f(x)=\frac{1}{2}{x}^2-3$ .

Graph:

The given function is sketched as shown below.

  

Interpretation

From the graph, it can be seen that the given function is initially decreasing in nature over a range of values and then increases with increase in x .

(b)

To determine

To find: the nature of the function from the graph and using a table of values of the function.

Answer to Problem 33E

The given function decreases in the range $(-\infty ,0)$ and then increases in the range $(0,\infty )$ .

Explanation of Solution

Given information:

The function $f(x)=\frac{1}{2}{x}^2-3$ .

From the graph of the function in previous part, it can be seen that, for all values of x from -8 to 0, the value of f (x ) decreases. Then, in the range 0 to 8, it increases. Thus, $f(x)=\frac{1}{2}{x}^2-3$ is a decreasing function in the range $(-\infty ,0)$ and increasing function in the range (0, 8).

Now, the values of the given function for different values of x are shown in the below table.

  

From the table, it can be verified that, for all the values of x considered, the function initially shows a decreasing nature and the root point being (0,3) from where, the function shows a steady increase. The function is symmetric with respect to the y-axis and takes a parabolic path.