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?

1 Answer
Feb 12, 2018

Answer:

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}#