The values of two functions, f and g, are given in a table. One, both, or neither of them may be exponential. Give the exponential models for those that are. begin{array}{|l|l|l|}hline X&-2&-1&0&1&2hline f(x)&0.18&0.9&4.5&22.5&112.5hline g(x)&12&6&3&1.5&0.75hlineend{array} f(x)-? g(x)-?

Question
Exponential models
asked 2020-12-12
The values of two functions, f and g, are given in a table. One, both, or neither of them may be exponential. Give the exponential models for those that are.
\(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{l}\right|}{l}{\left|{l}\right|}\right\rbrace}{h}{l}\in{e}{X}&-{2}&-{1}&{0}&{1}&{2}\backslash{h}{l}\in{e}{f{{\left({x}\right)}}}&{0.18}&{0.9}&{4.5}&{22.5}&{112.5}\backslash{h}{l}\in{e}{g{{\left({x}\right)}}}&{12}&{6}&{3}&{1.5}&{0.75}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\)
f(x)-?
g(x)-?

Answers (1)

2020-12-13
The function f(x) with respect to x is given in the table :
\(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{l}\right|}{l}{\left|{l}\right|}\right\rbrace}{h}{l}\in{e}{X}&-{2}&-{1}&{0}&{1}&{2}\backslash{h}{l}\in{e}{f{{\left({x}\right)}}}&{0.18}&{0.9}&{4.5}&{22.5}&{112.5}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\)
From the table it is shown that ,
The multiplier for every next coordinate is 5.
Therefore the common ratio formed is ,
\(\displaystyle{r}={\frac{{{0.9}}}{{{0.18}}}}\)
\(\displaystyle={5}\)
The initial value of the function is \(\displaystyle{a}={4.5}\)
The general form of the exponential form is,
\(\displaystyle{y}={a}\cdot{r}^{{x}}\)
Substitute the values,
\(\displaystyle{y}={4.5}{\left({5}\right)}^{{x}}\)
Thus the exponential function formed is \(\displaystyle{f{{\left({x}\right)}}}={y}={4.5}{\left({5}\right)}^{{x}}\)
The function g(x) with respect to x is given in the table :
\(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{l}\right|}{l}{\left|{l}\right|}\right\rbrace}{h}{l}\in{e}{X}&-{2}&-{1}&{0}&{1}&{2}\backslash{h}{l}\in{e}{g{{\left({x}\right)}}}&{12}&{6}&{3}&{1.5}&{0.75}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\)
From the table it is shown that ,
The multiplier for every next coordinate is 0.5.
Therefore the common ratio formed is,
\(\displaystyle{r}={\frac{{{6}}}{{{12}}}}\)
\(\displaystyle={0.5}\)
The initial value of the function is 3.
The general form of the exponential form is,
\(\displaystyle{y}={a}\cdot{r}^{{x}}\)
Substitute the values,
\(\displaystyle{y}={3}{\left({0.5}\right)}^{{x}}\)
Thus the exponential function formed is \(\displaystyle{g{{\left({x}\right)}}}={3}{\left({0.5}\right)}^{{x}}\)
0

Relevant Questions

asked 2021-02-25
The values of two functions, f and g, are given in a table. One, both, or neither of them may be exponential. Give the exponential models for those that are.
f(x)-?
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\(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{l}\right|}{l}{\left|{l}\right|}\right\rbrace}{h}{l}\in{e}{X}&-{2}&-{1}&{0}&{1}&{2}\backslash{h}{l}\in{e}{f{{\left({x}\right)}}}&{1.125}&{2.25}&{4.5}&{9}&{18}\backslash{h}{l}\in{e}{g{{\left({x}\right)}}}&{16}&{8}&{4}&{2}&{1}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\)
asked 2021-02-18
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asked 2020-11-08
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\(\begin{array}{|l|l|l|}\hline t(\text{in years})&m(\text{amoun of radioactive material})\\\hline0&\\\hline5730\\\hline11460\\\hline17190\\\hline\end{array}\)
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asked 2021-02-12
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a. Complete the table below. Make sure you justify your answer by showing all the steps.
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asked 2020-11-08
The following table lists the reported number of cases of infants born in the United States with HIV in recent years because their mother was infected.
Source:
Centers for Disease Control and Prevention.
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asked 2021-02-11
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asked 2021-01-31
The table gives the midyear population of Japan, in thousands, from 1960 to 2010.
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Use a calculator to fit both an exponential function and a logistic function to these data. Graph the data points and both functions, and comment on the accuracy of the models. [Hint: Subtract 94,000 from each of the population figures. Then, after obtaining a model from your calculator, add 94,000 to get your final model. It might be helpful to choose \(\displaystyle{t}={0}\) to correspond to 1960 or 1980.]
asked 2021-01-19
The annual sales S (in millions of dollars) for the Perrigo Company from 2004 through 2010 are shown in the table. \(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\text{Year}&{2004}&{2005}&{2006}&{2007}&{2008}&{2009}&{2010}\backslash{h}{l}\in{e}\text{Sales, S}&{898.2}&{1024.1}&{1366.8}&{1447.4}&{1822.1}&{2006.9}&{2268.9}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\) a) Use a graphing utility to create a scatter plot of the data. Let t represent the year, with \(\displaystyle{t}={4}\) corresponding to 2004. b) Use the regression feature of the graphing utility to find an exponential model for the data. Use the Inverse Property \(\displaystyle{b}={e}^{{{\ln{\ }}{b}}}\) to rewrite the model as an exponential model in base e. c) Use the regression feature of the graphing utility to find a logarithmic model for the data. d) Use the exponential model in base e and the logarithmic model to predict sales in 2011. It is projected that sales in 2011 will be $2740 million. Do the predictions from the two models agree with this projection? Explain.
asked 2020-11-30
The table gives the midyear population of Norway, in thousands, from 1960 to 2010.
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Use a calculator to fit both an exponential function and a logistic function to these data. Graph the data points and both functions, and comment on the accuracy of the models. [Hint: Subtract 3500 from each of the population figures. Then, after obtaining a model from your calculator, add 3500 to get your final model. It might be helpful to choose \(\displaystyle{t}={0}\) to correspond to 1960.]
asked 2020-12-24
The following table shows the approximate average household income in the United States in 1990, 1995, and 2003. (\(\displaystyle{t}={0}\) represents 1990.)
\(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\text{t(Year)}&{0}&{5}&{13}\backslash{h}{l}\in{e}\text{H(Household Income in}\ \${1},{000}{)}&{30}&{35}&{43}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\)
Which of the following kinds of models would best fit the given data?
Explain your choice of model. ( a, b, c, and m are constants.)
a) Linear: \(\displaystyle{H}{\left({t}\right)}={m}{b}\ +\ {b}\)
b) Quadratic: \(\displaystyle{H}{\left({t}\right)}={a}{t}^{{{2}}}\ +\ {b}{t}\ +\ {c}\)
c) Exponential: \(\displaystyle{H}{\left({t}\right)}={A}{b}^{{{t}}}\)
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