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}^{{{x}-{4}}} in the interval notation. c)The range of the function g{{left({x}right)}}={2}^{{{x}-{4}}} in the interval notation. d)The equation of the asymptote of the function g{{left({x}right)}}={2}^{{{x}-{4}}}.

Question
Transformation properties
asked 2021-03-12
Complete the tasks to determine:
a)To graf: The function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}.\)
b)The domain of the function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\) in the interval notation.
c)The range of the function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\) in the interval notation.
d)The equation of the asymptote of the function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}.\)

Answers (1)

2021-03-13
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,
image
image
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
\(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\)
image
Refer to the above table and draw the graph of the original function \(g{{\left({x}\right)}}={2}^{{{x}-{4}}}\)
image
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
\(y = 0\).
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Relevant Questions

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An automobile tire manufacturer collected the data in the table relating tire pressure x​ (in pounds per square​ inch) and mileage​ (in thousands of​ miles). A mathematical model for the data is given by
\(\displaystyle​ f{{\left({x}\right)}}=-{0.554}{x}^{2}+{35.5}{x}-{514}.\)
\(\begin{array}{|c|c|} \hline x & Mileage \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}\)
​(A) Complete the table below.
\(\begin{array}{|c|c|} \hline x & Mileage & f(x) \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}\)
​(Round to one decimal place as​ needed.)
\(A. 20602060xf(x)\)
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,45), (30,51), (32,56), (34,50), and (36,46). A parabola opens downward and passes through the points (28,45.7), (30,52.4), (32,54.7), (34,52.6), and (36,46.0). All points are approximate.
\(B. 20602060xf(x)\)
Acoordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2.
Data points are plotted at (43,30), (45,36), (47,41), (49,35), and (51,31). A parabola opens downward and passes through the points (43,30.7), (45,37.4), (47,39.7), (49,37.6), and (51,31). All points are approximate.
\(C. 20602060xf(x)\)
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (43,45), (45,51), (47,56), (49,50), and (51,46). A parabola opens downward and passes through the points (43,45.7), (45,52.4), (47,54.7), (49,52.6), and (51,46.0). All points are approximate.
\(D.20602060xf(x)\)
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,30), (30,36), (32,41), (34,35), and (36,31). A parabola opens downward and passes through the points (28,30.7), (30,37.4), (32,39.7), (34,37.6), and (36,31). All points are approximate.
​(C) Use the modeling function​ f(x) to estimate the mileage for a tire pressure of 29
\(\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.\) and for 35
\(\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.\)
The mileage for the tire pressure \(\displaystyle{29}\frac{{{l}{b}{s}}}{{{s}{q}}}\in.\) is
The mileage for the tire pressure \(\displaystyle{35}\frac{{{l}{b}{s}}}{{{s}{q}}}\) in. is
(Round to two decimal places as​ needed.)
(D) Write a brief description of the relationship between tire pressure and mileage.
A. As tire pressure​ increases, mileage decreases to a minimum at a certain tire​ pressure, then begins to increase.
B. As tire pressure​ increases, mileage decreases.
C. As tire pressure​ increases, mileage increases to a maximum at a certain tire​ pressure, then begins to decrease.
D. As tire pressure​ increases, mileage increases.
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To determine the solution of the initial value problem
\({y}{''}+{4}{y}= \sin{{t}}+{u}_{\pi}{\left({t}\right)} \sin{{\left({t}-\pi\right)}}:\)
\(y(0) = 0,\)
\(y'(0) = 0.\)
Also, draw the graphs of the solution and of the forcing function and explain the relation between the solution and the forcing function..
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