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James has decided to start saving for a cruise over the summer. His money is currently in the local bank, modeled by the function [tex]s(x) = 102[/tex]. He is able to do work around the neighborhood to earn extra money, modeled by the function [tex]a(x) = 8(x + 2)[/tex], where [tex]x[/tex] is measured in hours.

Explain to James how he can create a function that combines the two, and describe any simplification that can be done. (10 points)

Sagot :

GinnyAnsweridn
Certainly! Let's walk through how to help James create a function that combines his initial savings and his earnings from working, step by step.

### Step 1: Understand the Given Functions

James currently has two pieces of information:

1. Initial Savings in the Bank: This is modeled by the function \( s(x) = 102 \). This function indicates that, regardless of the value of \( x \), James has a flat amount of 102 dollars in the bank.

2. Earnings from Work: This is modeled by the function \( a(x) = 8(x + 2) \). This function indicates how much extra money James earns by working for \( x \) hours. For each hour worked, James earns 8 times the quantity of the hours plus 2.

### Step 2: Create a Combined Function

To find out James' total money after considering both his initial savings and his earnings from work, we need to combine the two functions.

- The combined function will be the sum of \( s(x) \) and \( a(x) \).

[tex]\[ \text{Total Savings} = s(x) + a(x) \][/tex]

### Step 3: Substitute the Given Functions

Let’s substitute \( s(x) \) and \( a(x) \) into this equation:

[tex]\[ \text{Total Savings} = 102 + 8(x + 2) \][/tex]

### Step 4: Simplify the Equation

We can simplify the combined function step by step.

- First, distribute the 8 in the \( a(x) \) function:

[tex]\[ 8(x + 2) = 8x + 16 \][/tex]

- Next, add this result to the initial savings:

[tex]\[ \text{Total Savings} = 102 + 8x + 16 \][/tex]

- Finally, combine the constant terms (102 and 16):

[tex]\[ \text{Total Savings} = 8x + 118 \][/tex]

So, the simplified combined function is:

[tex]\[ f(x) = 8x + 118 \][/tex]

### Step 5: Interpret the Combined Function

The function \( f(x) = 8x + 118 \) represents James' total savings after working for \( x \) hours. Here’s what each term represents:

- \( 8x \): This term shows the money James earns from working \( x \) hours at the rate of 8 dollars per hour after accounting for the additional 2 hours.
- \( 118 \): This is the sum of his initial savings (102 dollars) and the fixed amount calculated from his work earnings (16 dollars, which comes from \( 8 \times 2 \)).

This means that for:

- 0 hours of work: \( f(0) = 8(0) + 118 = 118 \)
- 1 hour of work: \( f(1) = 8(1) + 118 = 126 \)
- 2 hours of work: \( f(2) = 8(2) + 118 = 134 \)

Therefore, James can use the function [tex]\( f(x) = 8x + 118 \)[/tex] to model his total savings considering both his initial amount and his earnings from working [tex]\( x \)[/tex] hours.
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