Question

(a) Write an equation of the exponential function with base \(a > 0\).

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

(c) Find the range of the function \(y = {a^x}\)if \(a \ne 1\).

(d)

(i) Sketch the graph of the exponential function \(y = {a^x}\) if \(a > 1\).

(ii) Sketch the graph of the exponential function \(y = {a^x}\)if \(a = 1\).

(iii) Sketch the graph of the exponential function \(y = {a^x}\)if \(0 < a < 1\).

Short answer

(a) An equation of the exponential function with base \(a > 0\)is \(y = {a^x}\).

(b) The domain of the function \(y = {a^x}\)is \(( - \infty ,\infty )\).

(c) The range of the function\(y = {a^x}\), if\(a \ne 1\) is \((0,\infty )\).

(d) (i) The graph is monotonically increasing.

(ii) The graph is parallel to \(x\)-axis as the function is constant.

(iii) The graph is monotonically decreasing.

Step-by-step solution

Given data

The equation of the exponential function with base \(a > 0,\; \ne 1\).

Concept of domain of the function

In general, the domain of the function is the set of all independent value that defines that function.

The domain of an exponential functions\(y = {a^x},\;a > 0\)and\(a \ne 1\)is the set of real values.

That is,\( - \infty < x < \infty \).

Find an equation of the exponential function with base \(a > 0\)

(a)

The equation of the exponential function with base \(a > 0,\; \ne 1\)is \(y = {a^x}\).

Notice that this function is not defined when \(a < 0\).

Find the domain of the exponential function

(b)

In general, the domain of the function is the set of all independent value that defines that function.

The domain of an exponential functions \(y = {a^x},a > 0\) and \(a \ne 1\) is the set of real values.

That is, \( - \infty < x < \infty \).

Thus, the domain \(y = {a^x},a > 0\) and \(a \ne 1\) of is \(( - \infty ,\infty )\).

Find the range of the exponential function

(c)

In general, the range of the function is the set of all dependent values that defines that function.

The range of the exponential function, \(y = {a^x},\;a > 0\) and \(a \ne 1\) is the set of all positive \(y\)-values.

Since, \(a > 0\), the output cannot take negative values.

Thus, the range of the exponential function \(y = {a^x},\;a > 0\) and \(a \ne 1\)is \((0,\infty )\).

Sketch the graph of the exponential function \(y = {a^x}\)if \(a > 1\)

(d)

(i)

Obtain the graph of the exponential function \(y = {a^x},\;a > 1\) as shown below in figure 1

Figure 1

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

Sketch the graph of the exponential function \(y = {a^x}\)if \(a = 1\)

(ii)

Obtain the graph of the exponential function \(y = {a^x},\;a = 1\) as shown below in figure 2.

Figure 2

From figure 2, it is observed that the graph is parallel to \(x\)-axis as the function is constant.

Sketch the graph of the exponential function \(y = {a^x}\)if \(0 < a < 1\)

(iii)

Obtain the graph of the exponential function \(y = {a^x},\;0 < a < 1\) as shown below in figure 3.

Figure 3

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