Question

An interior designer wants to decorate a newly constructed house. The function f(x) = 49x^2 - 200 represents the amount of money he earns per room decorated, where x represents the number of rooms of he designs. The function g(x) = 1/7x represents the number of rooms the interior designer decorates, where x is the number of hours he works.
Part A: Determine the amount of money the interior designer will make decorating the house as a function of the hours he works.

Part B: Determine an expression to represent the difference quotient for the function found in Part A.

Answer 1

The given function for the amount of money earned is a function of the

function for the number of rooms decorated.

The correct responses are;

Part A: The amount made as a function hours worked, x, is; f(x) = x² - 200

Part B: The difference quotient f(x) = x² - 200 is 2·x + h

Reasons:

The given parameters are;

f(x) = 49·x² - 200 = Amount of money earned by the designers

Where;

x = The number of rooms decorated

$g(x) = \dfrac{1}{7} \cdot x$ = The number of rooms decorated

Where;

x = The number of hours he works

Part A: To determine the amount of money the interior designer will make

decorating the house as a function of the hours he works.

Solution:

The amount made as a function of the number of hours worked is given by

plugging in the number of rooms decorated, x = g(x) = (1/7)·x, in f(x) as

follows;

$f \left( \dfrac{1}{7} \cdot x \right) = 49 \cdot \left( \dfrac{1}{7} \cdot x \right)^2 - 200 = x^2 - 200$

Therefore, the amount of money the interior designer will make decorating

the house as a function of the hours he works, x, is f(x) = x² - 200.

Part B:

The difference quotient is given as follows;

$Difference \ quotient = \dfrac{f(x + h) - f(x)}{(x + h) - x} = \dfrac{f(x + h) - f(x)}{h}$

$\dfrac{f(x + h) - f(x)}{h} = \dfrac{((x + h)^2 - 200)- (x^2 - 200)}{h} = \dfrac{h^2 + 2 \cdot x \cdot h}{h} = h + 2 \cdot x$

The difference quotient for f(x) = x² - 200 is h + 2·x

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