A certain county has a tax code, where 10% tax is paid on all income up to the first
$10,000, a 15% tax is paid for any income over $10,000 and up to $25,000, and a tax
rate of 25% is paid on all income over $25,000. Find the piecewise function to
calculate the total tax T(x) on an income of x dollars.

The piecewise function to calculate the total tax T(x) on an income of x dollars will be T(x) = 0.1x, if x ≤ 10, 000, T(x) = 0.15x, if 10, 000 < x ≤ 25, 000, T(x) = 0.25x, if x > 25, 000.

What is piecewise function ?

Piecewise function is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain.

We have,

1. 10% tax is paid on all income up to the first $10,000.

2. 15% tax is paid for any income over $10,000 and up to $25,000.

3. Tax rate of 25% is paid on all income over $25,000.

Now,

For x ≤ 10,000 the rate is 10% i.e. 0.1,

So T(x) = 0.1x on this interval.

Now,

For 10, 000 < x ≤ 25, 000,

The rate is 15% i.e. 0.15.

Now,

For 25, 000 < x ,

The rate is 25% i.e. 0.25.

So,

Now,

When x = 10, 000, then y = 0.1 * 10, 000 = 1, 000.

Similarly,

For, 15% tax rate,

T(x) = 0.15(x − 10, 000) + 1, 000

T(x) = 0.15x − 1500 + 1000

We get,

T(x) = 0.15x − 500

Similarly

For tax rate of 25%, we get when x > 25, 000,

T(x) = 0.25x − 3000

So,

The piecewise function to calculate the total tax T(x) on an income of x dollars are,

T(x) = 0.1x, if x ≤ 10, 000

T(x) = 0.15x, if 10, 000 < x ≤ 25, 000

T(x) = 0.25x, if x > 25, 000

Hence, we can say that the piecewise function to calculate the total tax T(x) on an income of x dollars will be T(x) = 0.1x, if x ≤ 10, 000, T(x) = 0.15x, if 10, 000 < x ≤ 25, 000, T(x) = 0.25x, if x > 25, 000.

To know more about piecewise function click here

https://brainly.com/question/12561612

#SPJ2