Chapter 1.1, Problem 32E

To determine

To find: find the domain of the given function.

Answer to Problem 32E

The domain of the function is, $\left[-2,3\right]$

Explanation of Solution

Given information:

The given function is,

  $g\left(t\right)=\sqrt{3-t}-\sqrt{2+t}$

Concept used:

To represent the function:

A function $f$ is a rule that assigns to each element $x$ in a set $D$ exactly one element called $f\left(x\right)$ in a set E. The set $D$ and $E$ are sets of real numbers. The set $D$ is called the domain of the function.

If $f$ is a function with domain $D$ .

Then the set of ordered pairs are.

  $\left\{\left(x,f\left(x\right)\right)|x\in D\right\}$

The given function is

  $g\left(t\right)=\sqrt{3-t}-\sqrt{2+t}$

The square root a negative number is not defined (as a real number)

The domain of $g$ consists of all value of $t$ such that

  $\begin{array}{c}3-t\ge 0\\ t\le 3\end{array}$

And,

  $\begin{array}{c}2+t\ge 0\\ t\ge -2\end{array}$

Hence, the domain of $g$ is

  $-2\le t\le 3$

Therefore the domain of $g$ is the interval,

  $\left[-2,3\right]$