Complete the tasks to determine: a)To graf: The function g{{left({x}right)}}={2}^{{{x}-{4}}}. b)The domain of the function g{{left({x}right)}}={2}^{{{

CMIIh

CMIIh

Answered question

2021-03-12

Complete the tasks to determine:
a)To graf: The function g(x)=2x4.
b)The domain of the function g(x)=2x4 in the interval notation.
c)The range of the function g(x)=2x4 in the interval notation.
d)The equation of the asymptote of the function g(x)=2x4.

Answer & Explanation

Macsen Nixon

Macsen Nixon

Skilled2021-03-13Added 117 answers

a) Graph:
Use the transformation of the graph of the parent function y=2x to
graph the given function g(x)=2x4.
First, draw the graph of y=2x. For this purpose, draw the table for the
different points on the graph of the parent function y=2x.
Substitute - 1 for x in the above parent function y=2x.
y=21
=12
Now, substitute 0 for x .
y=20
=1
Finally, substitute 1 for x.
y=21
=2
Thus, the table for the graph of the parent function y=2x is,
image
image
Now, consider the original function g(x)=2x4. Use the properties of the transformation of the graph. In the graph of the function
g(x)=2x4,
the parent function graph y=2x is shifted to 4 units
positive towards the right.
Similarly, draw the table for the graph of the original function
g(x)=2x4
image
Refer to the above table and draw the graph of the original function g(x)=2x4
image
From the graph, it observed that function g(x)=2x4
is increasing in the domain (,).
The graph of the function g(x)=2x4 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(x)=2x4 in subpart (a).
From the graph, it can be concluded that the given function
g(x)=2x4 is defined for all the real values of x.
Thus, write the obtained domain in interval notation. (,)
Therefore, the domain of the given function is (,).
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(x)=2x4 in subpart (a).
From the graph, it can be concluded that the given function
g(x)=2x4 gives the output value above the y-axis.
Thus, write the obtained range in the interval notation.
(0,)
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(x)=2x4 in subpart (a).
From the graph, it can be concluded that the equation for the asymptote of the given function g(x)=2x4 is y=0.
Therefore, the equation for the asymptote of the given function is
y=0.

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