Word problem

[widmanns]

Suppose the government has been taxing each person's income at a marginal rate of 0.25 for every dollar in excess of $20,000. That is, the first $20,000 earned is not taxed. The government decides to generate extra revenue and wishes to avoid increasing the tax burden on low- or middle-income earners. Therefore, the government decides to impose a lump sum surtax of $1000 on every person who earns $60,000 or more.
a. Write and graph income after tax, y, as a function of income before tax, x.

So, as far as I understand this problem I am dealing here with a piecewise function. From 0 - $20,000 no tax, from $20,000 - $59,999 with 25% tax, and from $60,000 upwards with 25% tax plus a lump sum surtax of $1000.
However, I have no idea how can find the function to this problem in the end.

 

[Quaid]

Suppose the government has been taxing each person's income at a marginal rate of 0.25 for every dollar in excess of $20,000. That is, the first $20,000 earned is not taxed. The government decides to generate extra revenue and wishes to avoid increasing the tax burden on low- or middle-income earners. Therefore, the government decides to impose a lump sum surtax of $1000 on every person who earns $60,000 or more.

a. Write and graph income after tax, y, as a function of income before tax, x.

So, as far as I understand this problem I am dealing here with a piecewise function. From 0 - $20,000 no tax, from $20,000 - $59,999 with 25% tax, and from $60,000 upwards with 25% tax plus a lump sum surtax of $1000.

I have no idea how [to] find the function

Hi widmanns:

You explained that you have "no idea", but surely you have had some thoughts, regarding this function. There must be something about it that you understand.

I don't know where to begin helping. Are there confusing words, in the exercise statement? Do you know how to calculate symbolic percentages? Are you clear on the meaning of symbols x and y, in this exercise? Do you require examples of piecewise functions?

For example, where exactly are you getting stuck, when trying to write the expression for "income minus tax", for the first piece in the domain?

That is, when x is in the interval [0, 20000], it's fairly clear what y equals. Did you write that piece?

Please be as specific as you can. Cheers

 

[HallsofIvy]

Suppose the government has been taxing each person's income at a marginal rate of 0.25 for every dollar in excess of $20,000. That is, the first $20,000 earned is not taxed. The government decides to generate extra revenue and wishes to avoid increasing the tax burden on low- or middle-income earners. Therefore, the government decides to impose a lump sum surtax of $1000 on every person who earns $60,000 or more.
a. Write and graph income after tax, y, as a function of income before tax, x.

So, as far as I understand this problem I am dealing here with a piecewise function. From 0 - $20,000 no tax, from $20,000 - $59,999 with 25% tax, and from $60,000 upwards with 25% tax plus a lump sum surtax of $1000.
However, I have no idea how can find the function to this problem in the end.

The problem does NOT ask you to find a "function"- it gives you one. You are asked to graph the function you are given.
To do that, draw a horizontal axis marked from 0 to whatever top salary you want and vertical axis from 0 to whatever top tax you want. As you say, for salary from 0 to $20,000, there is no tax so the graph is just the horizontal axis itself. For salary from $20,000 to $60,000, the graph is a straight line passing through (20,000, 0) with slope 0.25. (Equivalently, since .25(60000)= 15000, it is the line through (20000, 0) and (60000, 15000).) For salary greater than $60,000, the graph is a straight line with the same slope but 1000 above. (Equivalently, .25(100000)+ 1000= 26000, it is the line through (60000, 16000) and (100000, 26000).

 

[wjm11]

Suppose the government has been taxing each person's income at a marginal rate of 0.25 for every dollar in excess of $20,000. That is, the first $20,000 earned is not taxed. The government decides to generate extra revenue and wishes to avoid increasing the tax burden on low- or middle-income earners. Therefore, the government decides to impose a lump sum surtax of $1000 on every person who earns $60,000 or more.
a. Write and graph income after tax, y, as a function of income before tax, x.

So, as far as I understand this problem I am dealing here with a piecewise function. From 0 - $20,000 no tax, from $20,000 - $59,999 with 25% tax, and from $60,000 upwards with 25% tax plus a lump sum surtax of $1000.
However, I have no idea how can find the function to this problem in the end.

Just break the function into the three parts you have described. It helps to graph them first. Here' is an example of a piecewise function with two parts, but the principle is the same:

http://www.purplemath.com/modules/graphing3.htm#piecewise

 

[Quaid]

The problem does NOT ask you to find a "function"- it gives you one. You are asked to graph the function you are given.

The instructions also specify to "write" the function. I believe that the OP intends "find" as "determine and write".

Also, I think you misread the function statement; the output is not tax.

 

[widmanns]

Thank you all so much for your input. It would be much appreciated if anyone of you could tell me if my function f(x)= { 20,000 (0 -20,000); 0.25x-500.25 (20,001-59.999), 0.25x-1500.25 (60,000< x) is correct?