Problem 100 - Precalculus Essentials

Question

A department store has two locations in a city. From 2012 through \(2016,\) the profits for each of the store's two branches are modeled by the functions \(f(x)=-0.44 x+13.62\) and \(g(x)=0.51 x+11.14 .\) In each model, \(x\) represents the number of years after \(2012,\) and \(f\) and \(g\) represent the profit, in millions of dollars. a. What is the slope of \(f ?\) Describe what this means. b. What is the slope of \(g\) ? Describe what this means. c. Find \(f+g .\) What is the slope of this function? What does this mean?

Step-by-Step Solution

Answer

a. The slope of f(x) is -0.44 which means the profit of the first branch decreases by 0.44 million dollars each year. b. The slope of g(x) is 0.51 which means the profit of the second branch increases by 0.51 million dollars each year. c. The function f+g is 0.07x + 24.76 and its slope is 0.07, meaning the combined profit of both branches increases by 0.07 million dollars each year.

Step 1: Finding slope of f(x)

Slope of f(x) can be found as the coefficient of x in the function f(x)=-0.44x+13.62. Here, the slope of f(x) is -0.44. This means for each additional year after 2012, the profit of the first branch decreases by 0.44 million dollars.

Step 2: Finding slope of g(x)

Slope of g(x) can be obtained as the coefficient of x in the function g(x)=0.51x+11.14. Here, the slope of g(x) is 0.51. This means for each additional year after 2012, the profit of the second branch increases by 0.51 million dollars.

Step 3: Finding f+g and slope of f+g

The sum of f(x) and g(x) is -(0.44)x + 13.62 + 0.51x + 11.14, simplifies to 0.07x + 24.76. The slope of this function is 0.07. This means for each additional year after 2012, the combined profit of both branches increases by 0.07 million dollars.