Finish the two parts of the piecewise function:

Rosa saved $100 to spend on vacation. For the first 3 days of her vacation she spent $20 each day. Then for the next 2 days, she spent nothing. After those 5 days, she spent $10 each day until her savings were depleted.

View photo, I've done a lot of the work, just can't figure out the 1st and 3rd equations.

Rosa begins her vacation with $100 in savings. Over the first three days, she spends $20 each day. Therefore, her savings decrease by $20 per day. If we let [tex]x[/tex] represent the number of days since the start of her vacation, her savings after [tex]x[/tex] days can be calculated by subtracting $20 multiplied by [tex]x[/tex] from her initial $100. This gives us the function:

[tex]f(x) = 100 - 20x[/tex]

This applies for the days from 0 to 3 inclusive, so the interval is:

[tex]0 \leq x \leq 3[/tex]

[tex]\dotfill[/tex]

Day 4 to Day 5

On the fourth and fifth days of her vacation, Rosa does not spend any money.

As she spends $20 each day for the first 3 days, by the end of the third day, she has already spent $20 × 3 = $60, leaving her with:

[tex]\$100 - \$60 = \$40[/tex]

Since there is no spending on the fourth and fifth days, her savings remain constant at $40 during this period. Hence, the function representing her savings on these days is:

[tex]f(x) = 40[/tex]

This is valid for the interval:

[tex]3 < x \leq 5[/tex]

[tex]\dotfill[/tex]

Day 6 onwards

Starting from the sixth day, Rosa resumes spending, but now at a reduced rate of $10 per day.

At the beginning of the sixth day, she has $40. For each additional day beyond the fifth, she spends $10, reducing her savings by $10 each day.

To express the number of days since the end of the fifth day, we use [tex]x-5[/tex]. Therefore, her savings after [tex]x[/tex] days can be calculated by subtracting $10 multiplied by [tex]x-5[/tex] from $40:

[tex]f(x) = 40 - 10(x - 5)[/tex]

Simplify:

[tex]f(x) = 40 - 10x + 50 \\\\ f(x) = 90 - 10x[/tex]

This function applies for the interval:

[tex]5 < x\leq9[/tex]

[tex]\dotfill[/tex]

Complete Piecewise Function

Therefore, the complete piecewise function that describes Rosa's savings over the duration of her vacation is:

[tex]f(x) = \begin{cases} 100 - 20x\quad 0 \leq x \leq 3 \\40\qquad \qquad\: 3 < x \leq 5 \\90 - 10x \quad \:\:5 < x\leq9\end{cases}[/tex]