For Exercises 6–10, use the table on the previous page that...

For Exercises 6–10, use the table on the previous page that provides a historical look at Social Security and Medicare. Express the 1993 Social Security tax function as a piecewise function $t(x),$ where $x$ is the annual income. Graph the function. What are the coordinates of the cusps on the graph?

For Exercises 6–10, use the table on the previous page that provides a historical look at Social Security and Medicare.
Express the 1993 Social Security tax function as a piecewise function $t(x),$ where $x$ is the annual income. Graph the function. What are the coordinates of the cusps on the graph?

Key Concepts

Piecewise Functions

A piecewise function is a function that is defined by different expressions over various intervals of the independent variable. In many applied settings, such as modeling tax systems, these types of functions allow for different behaviors (or rates) to be expressed in different ranges, making them ideal for representing systems where rules change at specific thresholds.

Tax Bracket Systems

A tax bracket system divides the total income into segments and applies distinct tax rates to each segment. This structure inherently leads to a piecewise definition, as each bracket is governed by its own rate and possibly a cap on the amount of income taxed at that rate, reflecting the progressive nature of many taxation systems.

Graphing Piecewise Functions

Graphing piecewise functions involves plotting each individual segment according to its corresponding equation across its specified domain. This process requires careful attention at the boundaries where the equations change, ensuring that the graph accurately reflects any discontinuities or changes in the rate of change, which are pivotal in understanding how the overall function behaves.

Cusps in Graphs

Cusps are points on a graph where the function exhibits a sharp change in direction, typically where two different linear segments meet. In the context of tax functions, these points correspond to income thresholds at which the tax rate changes, making them critical in analyzing the behavior and structure of the taxation function.

Transcript

00:02 What i've got up on the screen is the information for social security tax in 1993.

00:08 So this rate for social security was 6 .2 percent, and the maximum amount of income you would pay social security on was $57 ,600.

00:17 So if you made $70 ,000, you only paid social security tax on the first $57 ,600.

00:25 So what we're going to do with this information is we're going to write this as a piecewise function.

00:31 We're going to write as t of x.

00:33 X is going to be the annual income, and then we're going to draw the graph.

00:37 So to write this function out, we're going to say t of x.

00:43 And the way we find the percentage, if you want to find 6 .2 % of something, you multiply the percentage in decimal form times the number.

00:52 So if x is my income, i'm going to do 0 .062 times x.

01:00 And this is only true if it's under $57 ,600.

01:04 Because for over that, i don't have to pay taxes on it.

01:07 So this is going to be if x is less than $57 ,600.

01:16 Now, if it's over that, like if you make $70 ,000, there's a maximum amount of social security you could pay.

01:23 The maximum amount is that's 6 .2 % of $57 ,600.

01:31 The way you find that is we multiply.

01:35 If you multiply that, you get $3 ,751 .20.

01:44 So if you make anything over $57 ,600, that's all you're going to pay.

01:51 So that's the second part of our piecewise function is going to be 3 ,751 .20...

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