Chapter 1.4, Problem 48E

Solution

To find: The domain of the function $\left(\frac{f}{g}\right)\left(x\right)$ .

The domain of the function $\left(\frac{f}{g}\right)\left(x\right)$ is equal to $\left(-\infty ,4\right)$ .

Given information:

The given two functions are $f\left(x\right)=x-7$ and $g\left(x\right)=\sqrt{4-x}$ .

Calculation:

The given functions are $f\left(x\right)=x-7$ and $g\left(x\right)=\sqrt{4-x}$ .

Calculate the quotient function $\left(\frac{f}{g}\right)\left(x\right)$ below:

  $\begin{array}{c}\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}\\ =\frac{x-7}{\sqrt{4-x}}\end{array}$

For the domain of the function $\left(\frac{f}{g}\right)\left(x\right)$ , the denominator should not be equal to zero.

Therefore,

  $\begin{array}{c}\sqrt{4-x}\ne 0\\ 4-x\ne 0\\ x\ne 4\end{array}$

Since function $\left(\frac{f}{g}\right)\left(x\right)$ consists of all numbers that belong to both the domain of $f$ and the domain of $g$ .

The function $g\left(x\right)=\sqrt{4-x}$ . So, the domain of $g\left(x\right)$ is below:

  $\begin{array}{c}4-x\ge 0\\ x\le 4\end{array}$

Now, the overall domain of the quotient function $\left(\frac{f}{g}\right)\left(x\right)$ is

  $x\in \left(-\infty ,4\right)$