# The function f that models the purchase price of the washing machine as a function of the sticker price x . ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071 ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 2.6, Problem 65E

(a)

To determine

## The function f that models the purchase price of the washing machine as a function of the sticker price x.

Expert Solution

The value of the function f is f(x)=0.9x .

### Explanation of Solution

Given:

The discount on the washing machines is 10%.

Calculation:

The sticker price of washing machine is x.

The discount applied on washing machine is 10%.

Since the function f is a function that models the purchase price of the washing machine as a function of the sticker price x, therefore the function f is written as the difference between sticker price and purchase price,

f(x)=x10100x=100x10x100=90x100=0.9x

Thus, the value of the function f is f(x)=0.9x .

(b)

To determine

### The function g that models the purchase price of the washing machine as a function of the sticker price x.

Expert Solution

The value of the function g is g(x)=x100 .

### Explanation of Solution

Given:

The rebate applied on washing machine is \$100 .

Calculation:

Since the function g is the function that models the purchase price of the washing machine as a function of the sticker price x, therefore the function g is written as the difference between sticker price and rebate amount,

g(x)=x100

Thus, the value of the function g is g(x)=x100 .

(c)

To determine

### The values of f∘g , g∘f , and the composition function which gives the better deal.

Expert Solution

The value of fg is (fg)(x)=0.9x90 and the value of gf is (gf)(x)=0.9x100 . The purchase price from the composite function gf gives the better deal.

### Explanation of Solution

Given:

From part (a), the value of function f is given below,

f(x)=0.9x (1)

From part (b), the value of function g is given below,

g(x)=x100 (2)

Calculation:

The composite function fg is expressed as,

(fg)(x)=f(g(x))

From equation (2), substitute x100 for g(x) in above expression,

(fg)(x)=f(x100) . (3)

Substitute x100 for x in equation (1), to find the value of f(x100) ,

f(x100)=0.9(x100)=0.9x90

Substitute 0.9x90 for f(x100) in equation (3), to find the value of fg ,

(fg)(x)=0.9x90

The purchase price from the composite function fg is 0.9x90 . It shows the purchase price when rebate applied first and then discount applied.

The composite function gf is expressed as,

(gf)(x)=g(f(x))

From equation (1), substitute 0.9x for f(x) in above expression,

(gf)(x)=g(0.9x) . (4)

Substitute 0.9x for x in equation (2), to find the value of g(0.9x) ,

g(0.9x)=(0.9x)100

Substitute 0.9x100 for g(0.9x) in equation (4), to find the value of gf ,

(gf)(x)=0.9x100

The purchase price from the composite function gf is 0.9x100 . It shows the purchase price when discount applied first and then rebate applied.

The purchase price from the composite function gf is less than the purchase price from the composite function fg .

Thus, the value of fg is (fg)(x)=0.9x90 and the value of gf is (gf)(x)=0.9x100 . The purchase price from the composite function gf gives the better deal.

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