Write a piecewise definition for the tax due T(x) on an income of x dollars
When x <= $15,000..... 3% of taxable income
When $15,000 < x <= $30,000....... $450 plus 5.85% of excess over $15,000
When x > $30,000........ $1327.50 plus 6.8% of excess over $30,000
I know the first function would be 0.03x, but I am having trouble with the 2nd two. Please help!
Thank you! Is there a way you could show me what the graph looks like? Or explain the lines/end axis points?
When x <= $15,000: T(x) = 0.03x
When $15,000 < x <= $30,000: T(x) = $450 + 0.0585(x - $15,000)
When x > $30,000: T(x) = $1,327.50 + 0.068(x - $30,000)
Now, let me put a humorous twist on these definitions:
When your income is less than or equal to $15,000, the tax due is just 3% of your taxable income. It's like a generous discount, except it's not really a discount - it's money you have to give away. Isn't the government great at math?
When your income is more than $15,000, but not exceeding $30,000, things get a bit trickier. You have to pay a fixed amount of $450, because why not add a random number to confuse you? On top of that, you owe 5.85% of the money you make above $15,000. Talk about getting taxed for success!
If you're fortunate enough to make more than $30,000, brace yourself for a bigger hit. You'll need to cough up $1,327.50 upfront, because the government wants to make sure you're serious about paying taxes. But wait, there's more! For every dollar you earn over $30,000, an extra 6.8% will be snatched away like a magician pulling a rabbit out of a hat, except it's your money disappearing. Abracadabra, poverty!
Remember, don't take my comedic interpretation as legal advice. The numbers are real, but the jokes are just me clowning around.
Sure! Let's break down the problem into two parts:
Part 1: When $15,000 < x <= $30,000
In this range, the tax due consists of two components:
1. A fixed amount of $450.
2. An additional 5.85% of the excess over $15,000.
To express this as a piecewise function, we need to consider two scenarios:
a) If the income (x) is less than or equal to $15,000, the tax due in this range is not applicable. Therefore, we can write it as 0.
b) If the income (x) is between $15,000 and $30,000:
- Subtract $15,000 from the income to calculate the excess above $15,000.
- Multiply the excess by 5.85% to calculate the additional tax.
- Add the fixed amount of $450 to the additional tax.
So the second part of the piecewise function can be written as:
T(x) = 0, for x <= $15,000
T(x) = $450 + (0.0585 * (x - $15,000)), for $15,000 < x <= $30,000
Part 2: When x > $30,000
In this range, the tax due consists of two components:
1. A fixed amount of $1327.50.
2. An additional 6.8% of the excess over $30,000.
Again, let's express this as a piecewise function:
T(x) = 0, for x <= $15,000
T(x) = $450 + (0.0585 * (x - $15,000)), for $15,000 < x <= $30,000
T(x) = $1327.50 + (0.068 * (x - $30,000)), for x > $30,000
So, combining both parts, the piecewise definition for the tax due T(x) on an income x can be written as:
T(x) = 0, for x <= $15,000
T(x) = $450 + (0.0585 * (x - $15,000)), for $15,000 < x <= $30,000
T(x) = $1327.50 + (0.068 * (x - $30,000)), for x > $30,000
Hope this helps! Let me know if you have any further questions.
and the 2nd one would be 450+0.0585(x-15,000) which I would simplify. I still need help on the last one!
T1(x) = .03x ------ you had that
T2(x) = 450 + .0585(x - 15000)
T3(x) = 1327.50 + .068(x-30000)