Chapter 5, Problem 55RE

(a)

To determine

To Find: The domain of the $f$ .

Answer to Problem 55RE

The domain of the function is $\left(-\infty ,\infty \right)$ .

Explanation of Solution

Given:

The given function is $f\left(x\right)=2^{x-3}$ .

Calculation:

Consider the given function is,

  $f\left(x\right)=2^{x-3}$

The function is defined for all the value of s $x$ , thus the domain of the function is $f\left(x\right)=\left(-\infty ,\infty \right)$ .

(b)

To determine

To Find: The graph for the function $f$ .

Explanation of Solution

Given:

The given function is $f\left(x\right)=2^{x-3}$ .

Calculation:

The graph for the given function is shown in Figure 1

  

Figure 1

(c)

To determine

To Find: The range and the asymptote of the graph.

Answer to Problem 55RE

The range of the function is, $\left(0,\infty \right)$ and the horizontal asymptotes of the given function is $y=0$ .

Explanation of Solution

Given:

The given function is $f\left(x\right)=2^{x-3}$ .

Calculation:

Consider the given function,

  $f\left(x\right)=2^{x-3}$

For the above function the range of the function lies above the x axis so the range of the function is

  $\left(0,\infty \right)$

The case when $x\to \infty$ , the given function approaches to zero and as $x\to -\infty$ , the given function rises rapidly.

Thus, the horizontal asymptotes of the given function is,

  $y=0$

(d)

To determine

To Find: The function $f^{-1}$ .

Answer to Problem 55RE

The inverse of the functions is $y={\log }_{2}x+3$ .

Explanation of Solution

Given:

The given function is $f\left(x\right)=2^{x-3}$ .

Calculation:

Consider the given function,

  $f\left(x\right)=2^{x-3}$

Then,

  $y=2^{x-3}$

Then, the inverse of the function is,

  $x=2^{y-3}$

Take the log of both the sides and solve as,

  $\begin{array}{l}\log x=\log 2^{y-3}\\ \log x=\left(y-3\right)\log 2\\ \frac{\log x}{\log 2}+3=y\\ y={\log }_{2}x+3\end{array}$

(e)

To determine

To Find: The range of the function.

Answer to Problem 55RE

The range of the function is $\left(-\infty ,\infty \right)$ .

Explanation of Solution

Given:

The given function is $f\left(x\right)=2^{x-3}$ .

Calculation:

Consider the given function,

  $f\left(x\right)=2^{x-3}$

The domain of the function $f\left(x^{-1}\right)$ is equal to the range of the $f\left(x\right)$ that is the set of all the real number $\left(-\infty ,\infty \right)$ and the range of the inverse function is the domain of the original function that is $\left(-\infty ,\infty \right)$ in the interval notation.

(f)

To determine

To Find: The graph of the function $f^{-1}$ .

Explanation of Solution

Given:

The inverse of the function is $y={\log }_{2}x+3$ .

Calculation:

The graph for the inverse of the function is shown in Figure 2

  

Figure 2