I'm looking for some help with this. I'm trying to understand the concepts of function but am having a hard time

Question

 For income x (in dollars), a particular state's income tax T (in dollars) is given by

 

t(x)={0.028x                if 0<=x<=2500

         {70+0.035(x-2500) if 2500<x<=7500

        {245+0.050(x-7500) if x>7500

 

(a) What is the tax on an income of $6,300? .

 

 

 

 

(b) What is the tax on an income of $63,000? 

Answer 1

As you present the problem, you have three different income ranges to consider and have three different equations to calculate the tax amount, defined in function notation as t(x):

 

1. Income "x" is from $0 to $2500:  use equation.....

t(x)=0.028x........................Eq1

 

2. Income "x" is from $2501 to $7500:  use equation......

t(x)=(70+0.035(x-2500)).....Eq2

 

3.  Income "x" is $7501 or greater:  use equation.....

t(x)=(245+0.050(x-7500))...Eq3

 

Part (a) of problem statement, x=$6300, satisfies parameters

        of #2, substitute x=6300 into Eq2...

t(6300)=(70+0.035(6300-2500))

           =(70+0.035(3800)

           =70+133

           =203

 

Part (b) of problem statement, x=$63,000, satisfies parameters 

        of #3, substitute x=63,000 into Eq3...

t(63,000)=(245+0.050(63,000-7500))

              =(245+0.050(55,500))

              =245+2775

              =3020

 

So tax on $6300 in part (a) is $203 and

tax on $63,000 in part (b) is $3020

 

Michael J. excellently identified this problem as a "piecewise function" in his answer.  It is a function that is defined by different rules over different parts of the domain. Attempt to identify the domain and range for each part, then attempt to graph the function.

 

 

Answer 2

well this is a simple mathematical question.

(a). Since 6,300 is between 2500 and 7500, we use the second equation.

Therefore,   

t(x)=70+0.035(x-2500), and the value for x is 6,300.

t(6,300)=70+0.035(6,300-2500)

t(6,300)=70+0.035*3,800

           =70+133=203.

Therefore, The the state tax from an income of 6,300 is equal to 203.

Answer 3

What we have here is a piecewise function.  For each range of x values, there is a different relationship.  This means that for each range of income, the income tax is calculated differently.

 

a)

 

x = 6300

 

This number is in between   2500  and  7500.

 

You will need to use the second function            70+0.035(x-2500)

to evaluate the tax.

 

 

 

b)

 

x = 63000

 

 

Use the last function   245+0.050(x-7500)

 

to evaluate the tax.

 

Answer 4

We need to plug in x=6300 into equation (2)

 

We end up getting

 

T (6300) = 70+0.035 (6300 - 2500)

 

= 70 + 0.035 (3800)

 

= 203

 

For part b do the same but substitute the value into the third equation.

I'm sorry it should be 203 not 163 as I had stated before.

 

Yes thanks for pointing it out Mr. Norbert appreciate that.