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Answer :
Let's go through the errors in Hoang's function and how it can be corrected step by step:
Original Function by Hoang:
[tex]\[ f(x)=\left\{
\begin{array}{ll}
30000 & 0 \leq x \leq 20 \\
40 & 3000060000
\end{array}
\right. \][/tex]
Errors Identified:
1. The first part of the function is incorrect because the checkup price should be a reasonable figure, not 30000. Also, the range [tex]\(0 \leq x \leq 20\)[/tex] seems to be mistakenly written as it's too small for annual income.
2. The second range condition mentions "30000 < x < 60000," but it seems the value 30000 isn't correct within this context.
3. The boundaries in each condition need clarification to reflect a logical relationship between income and price.
Correct Function:
1. Let's start by assuming the checkup pricing logic is as follows:
- For those with lower annual incomes (up to [tex]$20,000), they pay a certain amount.
- People earning more (from $[/tex]20,001 up to [tex]$60,000) pay a different amount.
- Those earning above $[/tex]60,000 pay the highest price.
2. The revised function is:
- If the annual income [tex]\(x\)[/tex] is between 0 and 20,000 (including 0 and 20,000), the checkup price is 300 dollars.
- If the annual income [tex]\(x\)[/tex] is between 20,001 and 60,000 (including 60,000), the checkup price is 40 dollars.
- If the annual income [tex]\(x\)[/tex] is greater than 60,000, the checkup price is 60 dollars.
3. Written correctly, it should look like:
[tex]\[ f(x)=\left\{
\begin{array}{ll}
300 & 0 \leq x \leq 20000 \\
40 & 20000 < x \leq 60000 \\
60 & x > 60000
\end{array}
\right. \][/tex]
Now, this function accurately describes a pricing model that considers income brackets, with decreasing prices for higher income ranges.
Original Function by Hoang:
[tex]\[ f(x)=\left\{
\begin{array}{ll}
30000 & 0 \leq x \leq 20 \\
40 & 3000060000
\end{array}
\right. \][/tex]
Errors Identified:
1. The first part of the function is incorrect because the checkup price should be a reasonable figure, not 30000. Also, the range [tex]\(0 \leq x \leq 20\)[/tex] seems to be mistakenly written as it's too small for annual income.
2. The second range condition mentions "30000 < x < 60000," but it seems the value 30000 isn't correct within this context.
3. The boundaries in each condition need clarification to reflect a logical relationship between income and price.
Correct Function:
1. Let's start by assuming the checkup pricing logic is as follows:
- For those with lower annual incomes (up to [tex]$20,000), they pay a certain amount.
- People earning more (from $[/tex]20,001 up to [tex]$60,000) pay a different amount.
- Those earning above $[/tex]60,000 pay the highest price.
2. The revised function is:
- If the annual income [tex]\(x\)[/tex] is between 0 and 20,000 (including 0 and 20,000), the checkup price is 300 dollars.
- If the annual income [tex]\(x\)[/tex] is between 20,001 and 60,000 (including 60,000), the checkup price is 40 dollars.
- If the annual income [tex]\(x\)[/tex] is greater than 60,000, the checkup price is 60 dollars.
3. Written correctly, it should look like:
[tex]\[ f(x)=\left\{
\begin{array}{ll}
300 & 0 \leq x \leq 20000 \\
40 & 20000 < x \leq 60000 \\
60 & x > 60000
\end{array}
\right. \][/tex]
Now, this function accurately describes a pricing model that considers income brackets, with decreasing prices for higher income ranges.
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