Chapter 1.5, Problem 5E

(a)

To determine

To write: An equation of the exponential function with base $b>0$.

Answer to Problem 5E

The equation that defines the exponential function is $y=b^x$.

Explanation of Solution

An exponential function is a function of the form $f\left(x\right)=a^x$, where a is a positive constant called its base.

The equation of the exponential function with base $b>0$ and $b\ne 1$ is $y=b^x$.

Notice that this function is not defined when $b<0$.

Therefore, the equation of the exponential function with base $b>0$ is $y=b^x$.

(b)

To determine

The domain of the exponential function obtained in part (a).

Answer to Problem 5E

The domain of the exponential function $y=b^x$ is $\left(-\infty ,\infty \right)$.

Explanation of Solution

The domain of a function is defined as the set of all possible values of the independent variable of the function for which the function is defined.

Consider the exponential function $y=b^x$, whose independent variable is x.

Since the function $y=b^x$ is defined for all values of x, its domain will be the set all real values, that is $\left(-\infty ,\infty \right)$.

Therefore, the domain of the exponential function $y=b^x$ is $\left(-\infty ,\infty \right)$.

(c)

To determine

The range of the function $y=b^x$ if $b\ne 1$.

Answer to Problem 5E

The range of the function $y=b^x$ if $b\ne 1$ is $\left(0,\infty \right)$.

Explanation of Solution

The range of a function is defined as the set of all possible values of the dependent variable of the function.

Consider the exponential function $y=b^x$, whose dependent variable is y.

Since $b>0\text{ and }b\ne 1$, the value of y is always positive.

So the range of the function is the set of all positive real values.

Therefore, the range of the function $y=b^x$ if $b\ne 1$ is $\left(0,\infty \right)$.

(d)

(i)

To determine

To sketch: The graph of the exponential function if $b>1$.

Explanation of Solution

The graph of the exponential function $y=b^x,b>1$ is shown below in Figure 1.

From Figure 1, it is observed that the graph is monotonically increasing.

(ii)

To determine

To sketch: The graph of the exponential function if $b=1$.

Explanation of Solution

The graph of the exponential function $y=b^x,b=1$ is shown below in Figure 2.

From Figure 2, it is observed that the graph is parallel to x-axis as the function is a constant.

(iii)

To determine

To sketch: The graph of the exponential function if $0<b<1$.

Explanation of Solution

The graph of the exponential function $y=b^x,0<b<1$ is shown below in Figure 3.

From Figure 3, it is observed that the graph is monotonically decreasing.