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In 1990 the Massachusetts Non-Resident State Income Tax calls for a tax of $5 \%$ on earned income and $10 \%$ on unearned income. Suppose a person has total income of $\$ 40,000 dollars of which amount $x$ is earned. Find her $\operatorname{tax}, t,$ as a function of $x$.

t=-0.05 x+4000

Algebra

Chapter 1

Functions and their Applications

Section 1

The Line

Functions

Oregon State University

McMaster University

Harvey Mudd College

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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for this problem, we're going to try to find ah, function. We're going to try to relate tax to the amount of income that that somebody is bringing in. So let's take a look at what we know. We know that there is a 5% tax on all earned income and there's a 10% tax on unearned income. Okay, so 5% unearned, 10% on unearned. So if we're gonna put together our tax, we know that that's going to be 5% of some amount, plus 10% of another amount. Well, we have a total. We know that this person has $40,000 in total income. What we don't know is how much is earned and how much is unearned. So we're going to use our variable X and X is going to tell us the amount of earned income. Okay, so that's what we know. Let's put together in equations my tax t of X. As a function of X, I have 5% which I'm going to write us a decimal 50.5 times the amount of earned income I have, which in this case is X. That's our variable. It's what we don't know. I also need to have 10% tax, which is 100.10 10% tax on the unearned. Well, that's going to be the total minus the earned. So that's going to give me my earned income tax and my unearned income tax. So now let's take a moment just pretty this up a little bit. Let's get rid of our parentheses. 10% of 40,000 is 4000 and then minus 1/10 times X and we can add are like terms, So that's gonna give you a negative 0.5 X plus 4000. So there is my function relating the amount of tax to the amount of earned and unearned income that we have.

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