Chapter 4.7, Problem 52PFA

a.

To determine

Write a function to find the first paycheck.

Answer to Problem 52PFA

  $f(x)=12x-36.$

Explanation of Solution

Given information: D’s new job pays $12 per hour .For the job, D must buy a,

Uniform for $36, this will be taken out of her first paycheck.

Formula used:

The equation of a line in point-slope form is $y=mx+b.$ where m is the slope and b is the y -intercept of the line.

Slop m is calculated by $m=\frac{y_2-y_1}{x_2-x_1}$ .

Calculation:

Let x be the number of hours D works and $f(x)$ be the amount of D’s first payback. Since D earns $12 per hour ,the slope is 12.D must pay $36 for D’s uniform so the y -intercept is -36 since her wages will be -36$ for 0 hours worked. Substitute m =12 and b =-36 into the slope-intercept form of a line to write the equation.

  $\begin{array}{l}f(x)=mx+b.\\ f(x)=12x-36.\end{array}$

b.

To determine

What do x and $f(x)$ represent in the context of this function.

Answer to Problem 52PFA

  $x$ represents the number of hours D work and $f(x)$ represents the amount of D’s paycheck for her first week.

Explanation of Solution

Given information: D’s new job pays $12 per hour .For the job, D must buy a,

Uniform for $36, this will be taken out of her first paycheck.

Formula used:

The equation of a line in point-slope form is $y=mx+b.$ where m is the slope and b is the y -intercept of the line.

Slop m is calculated by $m=\frac{y_2-y_1}{x_2-x_1}$ .

Calculation:

Let x be the number of hours D works and $f(x)$ be the amount of D’s first payback. Since D earns $12 per hour ,the slope is 12.D must pay $36 for D’s uniform so the y -intercept is -36 since her wages will be -36$ for 0 hours worked. Substitute m =12 and b =-36 into the slope-intercept form of a line to write the equation.

  $\begin{array}{l}f(x)=mx+b.\\ f(x)=12x-36.\end{array}$

The domain $x$ represents the number of hours D work and the range of $f(x)$ represents the amount of D’s paycheck for her first week.

c.

To determine

Write the inverse function.

Answer to Problem 52PFA

  $f^{-1}(x)=(1/12)x+3$ .

Explanation of Solution

Given information: : : D’s new job pays $12 per hour .For the job ,D must buy a,

Uniform for $36, this will be taken out of her first paycheck.

Formula used:

The equation of a line in point-slope form is $y=mx+b.$ where m is the slope and b is the y -intercept of the line.

Slop m is calculated by $m=\frac{y_2-y_1}{x_2-x_1}$ .

Calculation:

Let x be the number of hours D works and $f(x)$ be the amount of D’s first payback. Since D earns $12 per hour ,the slope is 12.D must pay $36 for D’s uniform so the y -intercept is -36 since her wages will be -36$ for 0 hours worked. Substitute m =12 and b =-36 into the slope-intercept form of a line to write the equation.

  $\begin{array}{l}f(x)=mx+b.\\ f(x)=12x-36.\\ y=12x-36.\\ \text{switch}x\text{and}y.\\ x=12y-36.\\ 12y=x+36\\ y=(1/12)x+36.\\ f^{-1}(x)=(1/2)x+36.\\ \end{array}$

d.

To determine

What do x and $f^{-1}(x)$ represent in the context of the inverse function..

Answer to Problem 52PFA

For the original function , the domain $x$ represents the number of hours D work and the range of $f(x)$ represents the amount of D’s paycheck for D’s first week .For the inverse function, the range of $f^{-1}(x)$ then represents the number of hours D works and the domain x represents the amount of D’s r paycheck for her first week.

Explanation of Solution

Given information: : D’s new job pays $12 per hour .For the job ,D must buy a,

Uniform for $36, this will be taken out of her first paycheck

Formula used:

The equation of a line in point-slope form is $y=mx+b.$ where m is the slope and b is the y -intercept of the line.

Slop m is calculated by $m=\frac{y_2-y_1}{x_2-x_1}$ .

Calculation:

Let x be the number of hours D works and $f(x)$ be the amount of D’s first payback. Since D earns $12 per hour ,the slope is 12.D must pay $36 for D’s uniform so the y -intercept is -36 since her wages will be -36$ for 0 hours worked. Substitute m =12 and b =-36 into the slope-intercept form of a line to write the equation.

  $\begin{array}{l}f(x)=mx+b.\\ f(x)=12x-36.\\ y=12x-36.\\ \text{switch}x\text{and}y.\\ x=12y-36.\\ 12y=x+36\\ y=(1/12)x+36.\\ f^{-1}(x)=(1/2)x+36.\\ \end{array}$

For the original function , the domain $x$ represents the number of hours D work and the range of $f(x)$ represents the amount of D’s paycheck for D’s first week .For the inverse function, the range of $f^{-1}(x)$ then represents the number of hours D works and the domain x represents the amount of D’s r paycheck for her first week.

e.

To determine

How much money does D have in her paycheck if D works 24 hours?..

Answer to Problem 52PFA

$252.

Explanation of Solution

Given information: D’s new job pays $12 per hour .For the job, D must buy a,

Uniform for $36, this will be taken out of her first paycheck.

Formula used:

The equation of a line in point-slope form is $y=mx+b.$ where m is the slope and b is the y -intercept of the line.

Slop m is calculated by $m=\frac{y_2-y_1}{x_2-x_1}$ .

Calculation:

Let x be the number of hours D works and $f(x)$ be the amount of D’s first payback. Since D earns $12 per hour ,the slope is 12.D must pay $36 for D’s uniform so the y -intercept is -36 since her wages will be -36$ for 0 hours worked. Substitute m =12 and b =-36 into the slope-intercept form of a line to write the equation.

  $\begin{array}{l}f(x)=mx+b.\\ f(x)=12x-36.\\ f(24)=12\times 24-36=252.\end{array}$

So, 252$, D have in her paycheck if D works 24 hours.

f.

To determine

How many hours would D have worked if the paycheck were for $132.00?..

Answer to Problem 52PFA

14 hours.

Explanation of Solution

Given information: D’s new job pays $12 per hour .For the job, D must buy a,

Uniform for $36, this will be taken out of D’s first paycheck.

Formula used:

The equation of a line in point-slope form is $y=mx+b.$ where m is the slope and b is the y -intercept of the line.

Slop m is calculated by $m=\frac{y_2-y_1}{x_2-x_1}$ .

Calculation:

Let x be the number of hours D works and $f(x)$ be the amount of D’s first payback. Since D earns $12 per hour ,the slope is 12.D must pay $36 for D’s uniform so the y -intercept is -36 since her wages will be -36$ for 0 hours worked. Substitute m =12 and b =-36 into the slope-intercept form of a line to write the equation.

  $\begin{array}{l}f(x)=mx+b.\\ f(x)=12x-36.\end{array}$

  $\begin{array}{l}{f}^{-1}(x)=(1/12)x+3\\ f^{-1}(132)=(1/12)\times 132+3=11+3=14.\end{array}$

So, 14 hours would D have worked if the paycheck were for $132.00..