Chapter 1, Problem 15RE

Solution

To find: The domain and range of a function.

The domain and rage of the function $g\left(x\right)=3\left|x\right|+8$ are $\left(-\infty ,\infty \right)$ and $\left[8,\infty \right)$ respectively.

Given information: The function is $g\left(x\right)=3\left|x\right|+8$ .

Method/Formula used:

Suppose $y=f\left(x\right)$ be a function $f:A\to B$ , then the set A is called the domain of function f and set B the co-domain. The range of the function is the set of points which are images of points of domain of function f . Thus, the range is the domain of the function $f^{-1}\left(x\right)$ . The figure below explains domain, co-domain and the range of f .

  

Calculations:

The given function is $g\left(x\right)=3\left|x\right|+8$ . Let $g:A\to B$

(a) Domain:

Clearly, the function $g\left(x\right)=3\left|x\right|+8$ is defined for all real numbers, therefore, the set of all real numbers $ℝ\text{or}\left(-\infty ,\infty \right)$ is the domain of function $g\left(x\right)=3\left|x\right|+8$ .

(b) Range:

Suppose $y=g\left(x\right)$ , then the set of all y for which there exists an ‘ x’ in the domain of the function of $g\left(x\right)$ or set of points y for which the inverse function $g^{-1}\left(y\right)=x$ is defined, is the range of $g\left(x\right)$ .

Now let,

  $\left.\begin{array}{c}y=3\left|x\right|+8\\ y-8=3\left|x\right|\\ x\Rightarrow \left[\begin{array}{c}\left(\frac{y-8}{3}\right),y>8\\ -\left(\frac{y-8}{3}\right),y<8\end{array}\right.\\ g^{-1}\left(y\right)\Rightarrow \left(\frac{y-8}{3}\right),y\ge 8\end{array}\right\}$ … (1)

From result (1), it is clear that $g^{-1}\left(y\right)$ for all real number greater than 8.

Hence, the range of the function $g\left(x\right)=3\left|x\right|+8$ is the set of real numbers greater than 8, i.e., the interval $\left[8,\infty \right)$ .

Thus, the domain and rage of the function $g\left(x\right)=3\left|x\right|+8$ are $\left(-\infty ,\infty \right)$ and $\left[8,\infty \right)$ respectively.