a) Graph:

Use the transformation of the graph of the parent function \({y}={2}^{x}\) to

graph the given function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}.\)

First, draw the graph of \({y}={2}^{x}.\) For this purpose, draw the table for the

different points on the graph of the parent function \({y}={2}^{x}.\)

Substitute - 1 for x in the above parent function \({y}={2}^{x}.\)

\({y}={2}^{ -{{1}}}\)

\(= \frac{1}{{2}}\)

Now, substitute 0 for x .

\({y}={2}^{0}\)

\(= 1\)

Finally, substitute 1 for x.

\({y}={2}^{1}\)

\(= 2\)

Thus, the table for the graph of the parent function \({y}={2}^{x}\) is,

Now, consider the original function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}.\) Use the properties of the transformation of the graph. In the graph of the function

\(g{{\left({x}\right)}}={2}^{{{x}-{4}}},\)

the parent function graph \({y}={2}^{x}\) is shifted to 4 units

positive towards the right.

Similarly, draw the table for the graph of the original function

\(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\)

Refer to the above table and draw the graph of the original function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\)

From the graph, it observed that function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\)

is increasing in the domain \((-\infty, \infty).\)

The graph of the function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\) shifts 4 units towards the right along the abscissa by using this property of transformation of graph.

b) The domain is the set of all the inputs for which the given function is defined and is real.

Refer to the graph of the given function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\) in subpart (a).

From the graph, it can be concluded that the given function

\(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\) is defined for all the real values of x.

Thus, write the obtained domain in interval notation. \((- \infty, \infty)\)

Therefore, the domain of the given function is (- \infty, \infty).

c)

The range of the given function is the set of all the dependent variables for which it is defined.

Refer to the graph of the given function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\) in subpart (a).

From the graph, it can be concluded that the given function

\(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\) gives the output value above the y-axis.

Thus, write the obtained range in the interval notation.

\((0, \infty)\)

Therefore, the range of the given function is \((0, pp).\)

(d)

An asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tend to infinity.

Refer to the graph of the function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\) in subpart (a).

From the graph, it can be concluded that the equation for the asymptote of the given function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}{i}{s}{y}={0}.\)

Therefore, the equation for the asymptote of the given function is

\(y = 0\).

Use the transformation of the graph of the parent function \({y}={2}^{x}\) to

graph the given function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}.\)

First, draw the graph of \({y}={2}^{x}.\) For this purpose, draw the table for the

different points on the graph of the parent function \({y}={2}^{x}.\)

Substitute - 1 for x in the above parent function \({y}={2}^{x}.\)

\({y}={2}^{ -{{1}}}\)

\(= \frac{1}{{2}}\)

Now, substitute 0 for x .

\({y}={2}^{0}\)

\(= 1\)

Finally, substitute 1 for x.

\({y}={2}^{1}\)

\(= 2\)

Thus, the table for the graph of the parent function \({y}={2}^{x}\) is,

Now, consider the original function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}.\) Use the properties of the transformation of the graph. In the graph of the function

\(g{{\left({x}\right)}}={2}^{{{x}-{4}}},\)

the parent function graph \({y}={2}^{x}\) is shifted to 4 units

positive towards the right.

Similarly, draw the table for the graph of the original function

\(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\)

Refer to the above table and draw the graph of the original function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\)

From the graph, it observed that function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\)

is increasing in the domain \((-\infty, \infty).\)

The graph of the function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\) shifts 4 units towards the right along the abscissa by using this property of transformation of graph.

b) The domain is the set of all the inputs for which the given function is defined and is real.

Refer to the graph of the given function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\) in subpart (a).

From the graph, it can be concluded that the given function

\(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\) is defined for all the real values of x.

Thus, write the obtained domain in interval notation. \((- \infty, \infty)\)

Therefore, the domain of the given function is (- \infty, \infty).

c)

The range of the given function is the set of all the dependent variables for which it is defined.

Refer to the graph of the given function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\) in subpart (a).

From the graph, it can be concluded that the given function

\(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\) gives the output value above the y-axis.

Thus, write the obtained range in the interval notation.

\((0, \infty)\)

Therefore, the range of the given function is \((0, pp).\)

(d)

An asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tend to infinity.

Refer to the graph of the function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\) in subpart (a).

From the graph, it can be concluded that the equation for the asymptote of the given function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}{i}{s}{y}={0}.\)

Therefore, the equation for the asymptote of the given function is

\(y = 0\).