Question

1. two friends, frank and grace, want to plan a trip to disney world. they both started with savings and have summer jobs that pay them by the hour. the function that represents franks savings is ( f(x)=11x + 350 ) and the function that represents graces savings is ( g(x)=15.25x + 75 ).

a. what does ( f(x)=g(x) ) mean in this context?
b. what does ( f(x)+g(x) ) mean in this context?
c. write the new function, ( h(x) ), that represents frank and graces savings combined.
d. what is the ( y )-intercept of the new function, ( h(x) ), and what does it mean in this context?

Explanation:

• a. In the context of functions \(f(x)\) (Frank's savings) and \(g(x)\) (Grace's savings), \(f(x)=g(x)\) means that the amount of money Frank has saved is equal to the amount of money Grace has saved after \(x\) hours of work.
• b. \(f(x)+g(x)\) represents the total amount of money that Frank and Grace have saved together after \(x\) hours of work.
• c. To find \(h(x)=f(x)+g(x)\), we substitute the given functions:

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• d. The function \(h(x)=26.25x + 425\) is in the slope - intercept form \(y = mx + b\), where \(b\) is the \(y\) - intercept. For \(h(x)\), the \(y\) - intercept is \(425\). In this context, it means that when \(x = 0\) (i.e., before they start working), the combined amount of money Frank and Grace have saved is \(\$425\).

Answer:

a. Frank's savings equal Grace's savings after \(x\) hours of work.
b. The total savings of Frank and Grace after \(x\) hours of work.
c. \(h(x)=26.25x + 425\)
d. The \(y\) - intercept is \(425\), meaning their combined initial savings (before working) is \(\$425\).