Chapter 7.6, Problem 41PPS

To determine

By using the graph of the given function $f\left(x\right)={\left(\frac{1}{2}\right)}^x$ write the value of $n$ for its given equation of $g\left(x\right)$ .

Answer to Problem 41PPS

The value of $n$ is $n=-3$

So, the equation of $g\left(x\right)=f\left(nx\right)$ becomes $g\left(x\right)=f\left(-3x\right)$

Explanation of Solution

Given information:

It is given in the question that, the parent function is $f\left(x\right)={\left(\frac{1}{2}\right)}^x$

And the graph of given function $g\left(x\right)=f\left(nx\right)$ is,

  

Since the graph of the function $f\left(x\right)={\left(\frac{1}{2}\right)}^x$ is

  

By observing the graph of the two given functions, the graph of $g\left(x\right)=f\left(nx\right)$ is horizontal stretch with the $factor-"3"$ and also show reflection with respect to $x-axis$ which is only possible when the value of $n$ is $n=-3$

So, the equation of $g\left(x\right)=f\left(nx\right)$ becomes $g\left(x\right)=f\left(-3x\right)$