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?

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Answer 1 · 2018-02-12T00:35:41

See a solution process below:

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 > $19000}$

A certain country taxes the first $19,000 of an individual&#39;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?