Chapter 7.6, Problem 74PFA

To determine

To identify the transformed function $g(x)$ which is the translation of the function $f(x)={\frac{1}{2}}^x$ to 5 units up and reflected across the y -axis.

Answer to Problem 74PFA

  $g(x)={\left(\frac{1}{2}\right)}^{-x}+5$

Explanation of Solution

Given:

Parent function: $f(x)={\left(\frac{1}{2}\right)}^x$

Given,

  $y=f(x)={\left(\frac{1}{2}\right)}^x$

Now, taking the reflection of the given function about y -axis, which means,

  $y=f(-x)={\left(\frac{1}{2}\right)}^{-x}$

Now, shift the obtained function 5 units up, which means,

  $y=f(-x)+5={\left(\frac{1}{2}\right)}^{-x}+5$

Thus, the above expression represents the required function: $g(x)=f(-x)+5={\left(\frac{1}{2}\right)}^{-x}+5.$

Conclusion:

Hence, the required function is: $g(x)={\left(\frac{1}{2}\right)}^{-x}+5.$