Question

An income tax follows the step-wise graph (where `x` is income): 0 for incomes 0 or less, `.14x` for `0<x<=10,000`, and `c+.21x` for `x>10,000`. What value of `c` will make the graph continuous?

Answer 1

`c=-700`

Explanation:

We have a function that is defined as:

`color(white)(0000)=0, color(white)(000000)x<=0`
`T(x)=0.14x, color(white)(00)0 < x <= 10,000`
`color(white)(0000)=c+.21x, x>10,000`

and we're looking for the value of `c` to make the graph continuous. (Either the second or third function needs to have a condition where it's equal to 0, so I've put it in the second equation. I've also assumed the second equation to be `.14x`)

To do that, we want the last possible value of `.14x` to be equal to the first value of `c+.21x` - this will join up the points and make the function continuous.

So we can write:

`.14(10000)=c+.21(10000)`

and solve for `c`:

`1400=c+2100`

`c=-700`

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As a practical point, tax rates generally are written in this manner:

0% on the first $10,000
10% on the next $20,000
20% on the next $40,000
35% on everything thereafter

And it gets calculated this way (with income of, say, $100,000):

`((Income, Tax Rate, Tax),("$10,000",0%,$0),("$20,000",10%,"$2,000"),("$40,000",20%,"$8,000"),(ul("$30,000"),35%,ul("$10,500")),("$100,000",color(white)(000),"$20,500"))`

With the bottom line being totals.