How Does Function Notation Work?

In calculus, we will need to  take a function (x) and write out (x+h) for that function. Let’s look at how to do this properly.

To do a problem like this, you need to understand exactly what the x in (x) represents and what the f represents. Let’s look at the function (x) = x2x. A function is a process. In this case, it is the process of

  1. Square the input
  2. Take the result and subtract the input

Notice that there is no mention of the x in the formula. That is because it is a placeholder representing the input. There is nothing special about x. We could have just as easily used a different letter as a placeholder for the input. If I had wanted to call the input t, I would have written

(t) = t2 – t

If the input had been represented by the word dog, I would have written

(dog) = dog2 – dog

The input variable is simply a placeholder…if a number is put in its place like 7, we get

(7) = 72 – 7 = 42

Notice that the process is the same. Square the input and subtract the input from the result. In this case, the input is 7 so we are squaring 7 and then subtracting 7 from the result.

Many students are confused by f(x+h). Now the input is represented by x+h instead of x. This means we need to square it and then subtract x+h from the result.

(x+h) = (x+h)2 – (x+h)

We can simplify this by foiling out the square,

(x+h) = (x+h)(x+h) = x2 +2xh + h2

And removing the parentheses after the subtraction we get

(x+h) =  x2 +2xh + h2xh

The handout below has more examples with this function.

How Do I Write Down a Piecewise Function For Postage?

In of April 2015, the US Postal Service established new postal rates for first class mail. The postage charged for first class mail is a function of its weight. The US Postal Service uses this table to describe the rates.


Problem Convert this table to a piecewise defined function that represents first class postage for letters weighing up to 3 ounces, using x as the weight in ounces and P as the postage in cents.

Solution This one always causes many questions. I suggest trying a bunch of different inputs (weights) and seeing how it works…then try to come up with the formula.

So, I created a table of possible weights. I made sure to include weights that were fractions of an ounce (something other than 1, 2, or 3). This allows us to understand what that phrasing means.

Now let us try to add in some corresponding postage amounts. Suppose a letter weighed 0.5 ounces. It would fall into that first part of the function “First ounce or fraction of an ounce”. So it would cost 49 cents. The same would be true of a letter weighing 0.75 or 1 ounce…both would cost 49 cents. We can add some numbers to the table.

In fact, any letter weighing 1 ounce of less (and greater than 0) would cost 49 cents.

Now what happens when the letter weighs a little more than 1 ounce? For a letter weighing 1.5 ounces, the first ounce would cost 49 cents and since the letter falls into another “additional ounce or fraction of an ounce”, the total cost would be  49 + 22 = 71. Any letter weighing more than 1 ounce up to 2 ounces would have the same exact cost, 71 cents. Now we can update our table:

As soon as we increase the weight to the next ounce, another 22 cents is added. So a letter weighing 2.5 or 3 ounces would cost 49 + 22 + 22 = 93.

The key thing to note is that for each ounce, the postage stays constant until the next ounce. The correct piecewise function needs to take this into account.

Start your piecewise function with where the pieces are valid:

Each piece corresponds to where the postage is constant and were the rates change. For instance, at x = 1, the postage changes from 41 to 71 cents since we have gone to a new ounce. In each weight interval, the postage is constant according to the table. This give us the function

A graph of the postage function P(x) looks like the one below.

Death and Piecewise Functions

Although this may seem a little gruesome, it is not uncommon for businesses to give discounts for volume sales. In this case, a mortician charges less per pound for bodies weighing more than a certain amount.

The local mortician charges by the pound for embalming according to the following table:

Find a piecewise linear function that models the cost as a function of weight.

Continue reading “Death and Piecewise Functions”

Piecewise Functions and Taxes

Most states in the US have some type of graduated income tax. This means that as your taxable income increases, the higher amounts of income are taxed at a higher and higher rate. There are several ways to express the relationship between taxable income and the corresponding amount of tax. In the state of Arizona, the instructions for Form 140 includes the table above for individuals filing single or married and filing single.

Continue reading “Piecewise Functions and Taxes”

Piecewise Functions and Taxes (Part 2)

In an another FAQ, I showed how to construct a piecewise function from a tax table on Arizona’s Form 140 Income Tax Form. Other states have descriptions of how income taxes are collected. In 2008, the following description was used to calculate state income taxes in Alabama.

  • For the first $500 in taxable income, the tax rate on that income is 2%.
  • For the next $2500 in taxable income, the tax rate on that income is 4%.
  • On all additional income, the tax rate on that income is 5%.

Let’s use this information to construct a piecewise function T(x) for the income tax as a function of the taxable income x.

Continue reading “Piecewise Functions and Taxes (Part 2)”