Question

A certain country taxes the first $19,000 of an individual's income at a rate of 15%, and all income over $19,000 is taxed at 25%. How do you find a piecewise-defined function T that specifies the total tax on an income of x dollars?

Answer 1

See a solution process below:

Explanation:

First, let's call the individual's income: `i`

Let's call the amount of tax they will pay: `t`

"Percent" or "%" means "out of 100" or "per 100",

Therefore, 15% can be written as `15/100`.

And, 25% can be written as `25/100`.

Then if, the individuals income is less than or equal to $19,000 the amount of tax they will pay in terms of `i` is:

`t = 15/100i`

`t = 3/20i`

The amount of tax they will pay if they earn more than $19,000 is:

`t = (15/100 xx $19000) + 25/100(i - $19000))`

`t = $2850 + 25/100(i - $19000))`

`t = $2850 + 25/100i - $4750`

`t = 25/100i - $1900`

`t = 1/4i - $1900`

The piece wise function would be:

`y = {3/20i" for "i <= $19000;`

`color(white)("..........")1/4i - $1900 " for "i > $19000}`