Question

# 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

Exponential models

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)-?
g(x)-??
$$\begin{array}{|l|l|l|}\hline X&-2&-1&0&1&2\\\hline f(x)&1.125&2.25&4.5&9&18\\\hline g(x)&16&8&4&2&1\\\hline\end{array}$$

2021-02-26
Given table show the value of the two functions f(x) and g(x) at -2,-1,0,1,2.
To find f(x):
From the table it is clear that for every increment of x by 1 unit, the value of f(x) gets multiplied by 2.
So, let $$\displaystyle{f{{\left({x}\right)}}}={k}{.2}^{{x}}$$.
Given $$\displaystyle{f{{\left({0}\right)}}}={4.5}\Rightarrow{k}{.2}^{{0}}={4.5}\Rightarrow{k}={4.5}$$
Hence $$\displaystyle{f{{\left({x}\right)}}}={\left({4.5}\right)}{.2}^{{x}}$$
To find g(x):
From the table it is clear that for every increment of x by 1 unit, the value of f(x) get halved.
So, let $$\displaystyle{g{{\left({x}\right)}}}={k}.{\left({\frac{{{1}}}{{{2}}}}\right)}^{{x}}$$
Given: $$\displaystyle{g{{\left({0}\right)}}}={4}\Rightarrow{k}.{\left({\frac{{{1}}}{{{2}}}}\right)}^{{0}}={4}\Rightarrow{k}={4}$$
Hence $$\displaystyle{g{{\left({x}\right)}}}={4}.{\left({\frac{{{1}}}{{{2}}}}\right)}^{{x}}$$
Thus, the exponential modes of two function are $$\displaystyle{f{{\left({x}\right)}}}={\left({4.5}\right)}{.2}^{{x}},\ {g{{\left({x}\right)}}}={4}.{\left({\frac{{{1}}}{{{2}}}}\right)}^{{x}}$$