Chapter 1.4, Problem 3E

Solution

To find: The formula and domain for functions $f+g$ $f-g$and $fg$ .

The required functions are: $\left(f+g\right)=\sqrt{x}+\sin x$ , $\left(f-g\right)=\sqrt{x}-\sin x$ and $\sqrt{x}\sin x$ . The domain of each function $\left(f+g\right)$ , $\left(f-g\right)$ and $fg$ is the set of real numbers $x\in \left(0,\infty \right)$ .

Given data: The functions $f\left(x\right)$ and $g\left(x\right)$ are $f\left(x\right)=\sqrt{x}$ ; $g\left(x\right)=\sin x$ .

Method/Formula used:

The sum, difference and product of the two functions $f\left(x\right)$ and $g\left(x\right)$ is also a function given by $f\left(x\right)+g\left(x\right)=\left(f+g\right)x$ , $f\left(x\right)-g\left(x\right)=\left(f-g\right)x$ and $f\left(x\right)\cdot g\left(x\right)=\left(f\cdot g\right)\left(x\right)$ . The domain of resulting functions $\left(f+g\right)x$ , $\left(f-g\right)x$ and $f\left(x\right)\cdot g\left(x\right)$ are the values of x that define these functions $\left(f+g\right)x$ $\left(f-g\right)x$ and $\left(f\cdot g\right)\left(x\right)$

Calculation:

The given functions are $f\left(x\right)=\sqrt{x}$ ; $g\left(x\right)=\sin x$ . Therefore, using the definition of $\left(f+g\right)\left(x\right)$ , $\left(f-g\right)\left(x\right)$ and $fg\left(x\right)$ are:

  $\left(f+g\right)\left(x\right)$ :

  $\begin{array}{c}\left(f+g\right)x=f\left(x\right)+g\left(x\right)\\ =\sqrt{x}+\sin x\end{array}$

The function $\sqrt{x}+\sin x$ is defined for all reals $x\ge 0$ .Therefore, the domain of $\left(f+g\right)x$ is $\left[0,\infty \right)$ .

  $\left(f-g\right)\left(x\right)$ :

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

The function $\sqrt{x}+\sin x$ is defined for all reals $x\ge 0$ .Therefore, the domain of $-x$ is $\left[0,\infty \right)$ .

  $\left(fg\right)\left(x\right)$ :

  $\left(fg\right)x=\sqrt{x}\sin x$

The function $\sqrt{x}\sin x$ is defined for all reals $x\ge 0$ .Therefore, the domain of $\left(fg\right)x$ is $\left[0,\infty \right)$ .

The graphs of functions $\sqrt{x}+\sin x$ , $\sqrt{x}-\sin x$ , and $\sqrt{x}\sin x$ are shown in Fig. (1), here that also shows that the functions $\sqrt{x}+\sin x$ , $\sqrt{x}-\sin x$ , and $\sqrt{x}\sin x$ are defined for all $x\ge 0$ , asserting that the domain of each of these function is $\left[0,\infty \right)$ .