Piecewise function problem for calculating income tax based on different income brackets.

It looks like you've uploaded a problem about applications of piecewise functions related to income tax, and there's also a "Your Scenario" section at the bottom with different tax brackets. Let's solve both parts carefully.

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🔹 Problem Statement (Original):

> In a certain country, income tax is assessed as follows:
> - No tax on income up to $15,000
> - Income over $15,000 and up to $30,000 is taxed at 12%
> - Income over $30,000 is taxed at 18%

We are asked to:

a) Write the piecewise function and sketch its graph
b) Determine how much tax is paid on an income of $17,000

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## ✔ Part (a): Write the Piecewise Function

Let’s define $ f(x) $ as the tax amount for an income of $ x $ dollars.

The tax is applied only to the portion of income in each bracket, not the entire income.

So we need to define $ f(x) $ piecewise based on income levels.

Step-by-step breakdown:

- For $ x \leq 15,000 $: No tax → $ f(x) = 0 $
- For $ 15,000 < x \leq 30,000 $: Tax only the amount above $15,000 at 12%
- So: $ f(x) = 0.12(x - 15,000) $
- For $ x > 30,000 $:
- First $15,000 → no tax
- Next $15,000 ($15,001 to $30,000) → taxed at 12%
- Amount over $30,000 → taxed at 18%
- So: $ f(x) = 0.12(30,000 - 15,000) + 0.18(x - 30,000) $
- Simplify: $ f(x) = 0.12(15,000) + 0.18(x - 30,000) = 1,800 + 0.18(x - 30,000) $

✔ Final Piecewise Function:

$$
f(x) =
\begin{cases}
0, & x \leq 15,000 \\
0.12(x - 15,000), & 15,000 < x \leq 30,000 \\
1,800 + 0.18(x - 30,000), & x > 30,000
\end{cases}
$$

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📈 Sketching the Graph:

- From $ x = 0 $ to $ x = 15,000 $: Flat line at $ f(x) = 0 $
- From $ x = 15,000 $ to $ x = 30,000 $: Straight line starting at (15,000, 0) with slope 0.12
- At $ x = 30,000 $: $ f(30,000) = 0.12(15,000) = 1,800 $
- For $ x > 30,000 $: Line starting at $ (30,000, 1,800) $ with steeper slope (0.18)

So the graph is:
- Horizontal from (0,0) to (15,000, 0)
- Then upward-sloping with slope 0.12 until (30,000, 1,800)
- Then upward-sloping with slope 0.18 beyond that

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## ✔ Part (b): Tax on $17,000

Since $ 15,000 < 17,000 \leq 30,000 $, use the second case:

$$
f(17,000) = 0.12(17,000 - 15,000) = 0.12 \times 2,000 = 240
$$

✔ Answer: $240 tax is due on $17,000 income.

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## ❗ But Wait — There's a "Your Scenario" at the Bottom!

You wrote:

> Your Scenario: There is no tax on income up to $12,000. Income over $12,000 and up to $25,000 is taxed at 10%. Income over $25,000 is taxed at 16%.

This appears to be a different problem — possibly your own version or a typo.

Let’s solve that one too, just in case it’s what you're supposed to answer.

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## 🔸 Your Scenario (Alternative Problem)

Tax rules:
- $ x \leq 12,000 $: $ f(x) = 0 $
- $ 12,000 < x \leq 25,000 $: taxed at 10% on amount over $12,000
- $ x > 25,000 $: taxed at 16% on amount over $25,000, plus previous taxes

✔ Piecewise Function:

$$
f(x) =
\begin{cases}
0, & x \leq 12,000 \\
0.10(x - 12,000), & 12,000 < x \leq 25,000 \\
0.10(25,000 - 12,000) + 0.16(x - 25,000), & x > 25,000
\end{cases}
$$

Simplify third case:
- $ 0.10 \times 13,000 = 1,300 $
- So: $ f(x) = 1,300 + 0.16(x - 25,000) $

Graph:

- Flat at 0 up to $12,000
- Slope 0.10 from $12,000 to $25,000 → ends at $ f(25,000) = 1,300 $
- Then slope 0.16 beyond $25,000

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## ✔ Now, if you want to find tax on $17,000 under your scenario:

$ 12,000 < 17,000 \leq 25,000 $ → use second case:

$$
f(17,000) = 0.10(17,000 - 12,000) = 0.10 \times 5,000 = 500
$$

✔ So under your scenario, tax on $17,000 is $500

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## 🟩 Final Answers

✔️ Original Problem (as stated):

- Piecewise function:
$$
f(x) =
\begin{cases}
0, & x \leq 15,000 \\
0.12(x - 15,000), & 15,000 < x \leq 30,000 \\
1,800 + 0.18(x - 30,000), & x > 30,000
\end{cases}
$$
- Tax on $17,000: $ \boxed{240} $

✔️ Your Scenario:

- Piecewise function:
$$
f(x) =
\begin{cases}
0, & x \leq 12,000 \\
0.10(x - 12,000), & 12,000 < x \leq 25,000 \\
1,300 + 0.16(x - 25,000), & x > 25,000
\end{cases}
$$
- Tax on $17,000: $ \boxed{500} $

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Let me know which one you were supposed to solve — but I’ve covered both!