Chapter 3.3, Problem 3MRPS

Solution

Which composition, $f(g\left(x\right))org(f\left(x\right))$ , represents your bonus if $x>100000$ ? Explain.

  $f\left(x\right)=x-100,000$

  $g\left(x\right)=0.03x$

  $g(f(x))$

Given:

You are working as a sales representative for a clothing manufacturer. You are paid an annual salary plus a bonus of 3% of your sales over $100000. Consider these two functions :

  $f\left(x\right)=x-100,000$

  $g\left(x\right)=0.03x$

Concept Used:

• The composition function $f\circ g$ can be found by replacing x by $g\left(x\right)$ in the function $f\left(x\right)$
• The domain of a function $y=f\left(x\right)$ is the set of all x values for which the function is defined.
• A rational function $y=f\left(x\right)$ is undefined for the values of x , for which the denominator is zero.
• A radical function is defined for positive values greater than or equal to zero.
• Zero product property: if $a\cdot b=0\text{then}a=0\text{or}b=0$

Calculation:

  $f\left(x\right)=x-100,000$

  $g\left(x\right)=0.03x$

  $\begin{array}{c}f(g(x))=f(0.03x)\\ =0.03x-100000\end{array}$

  $\begin{array}{c}g(f(x))=g(x-100000)\\ =0.03(x-100000)\\ =0.03x-3000\end{array}$

So $g(f(x))$ represents the bonus if $x>100000$ , because if $x>100000$ then putting in $f(g(x))$ it would be in negative that is bonus never comes in negative amount.