Chapter 1.2, Problem 17E

Solution

To find: the range of function.

The range is $(-\infty ,10]$ .

Given information: 

The function $f\left(x\right)=10-x^2$ .

Calculation:

Here,

  $x^2$ is zero when $x=0$ and positive all other times, since it's squared

The smallest value of the second term in the expression will be zero. All other values will be positive.

  $10\text{ -}$ any positive number gives a value less than $10$ .

  $10$ is the largest possible value of the function. As $x$ gets bigger, the function becomes a larger and larger negative number. A graph of the function would make that clearer.

  

Range is always looking from the bottom to the top along the $y-$ axis. Notice that the graph DOES exist coming from the bottom of the graph, but stops at $10.$ That's what makes the range $(-\infty ,10]$ .