Introduction to Functions Help

Introduction to Functions

A relation between two sets A and B is a collection of ordered pairs, where the first coordinate comes from A and the second comes from B . For example, if A = {1, 2, 3, 4} and B = {a, b, c} , one relation is the three pairs {(1, c ), (1, a ), (3, a )}. A function on sets A and B is a special kind of relation where every element of A is paired with exactly one element from B . The relation above fails to be a function in two ways. Not every element of A is paired with an element from B , 1 and 3 are used but not 2 and 4. Also, the element 1 is used twice , not once . There are no such restrictions on B ; that is, elements from B can be paired with elements from A many times or not at all. For example, {(1, a ), (2, a ), (3, b ), (4, b )} is a function from A to B.

Functions exist all around us. If a worker is paid by the hour, his weekly pay is a function of how many hours he worked. For any number of hours worked, there is exactly one pay amount that corresponds to that time. If A is the set of all triangles and B is the set of real numbers, then we have a function that pairs each triangle with exactly one real number that is its area. We will be concerned with functions from real numbers to real numbers. A will either be all of the real numbers or will be some part of the real numbers, and B will be the real numbers.

A linear function is one of the most basic kinds of functions. These functions have the form f ( x ) = mx + b . The only difference between f ( x ) = mx + b and y = mx + b is that y is replaced by f ( x ). Very often f ( x ) and y will be the same. The letter f is the name of the function. Other common names of functions are g and h . The notation f (x) is pronounced “ f of x ” or “ f at x .”

Evaluating a function at a quantity means to substitute the quantity for x (or whatever the variable is). For example, evaluating the function f (x) = 2 x − 5 at 6 means to substitute 6 for x.

f (6) = 2(6) − 5 = 7

We might also say f (6) = 7. The quantity inside the parentheses is x and the quantity on the right of the equal sign is y . One advantage to this notation is that we have both the x- and y -values without having to say anything about x and y . Functions that have no variables in them are called constant functions . All y -values for these functions are the same.

Examples

Find f (−2), f (0), and f (6) for .

We need to substitute −2, 0, and 6 for x in the function.
Find f (−8), f (π), and f (10) for f (x) = 16.

f (x) = 16 is a constant function, so the y -value is 16 no matter what quantity is in the parentheses.

f (−8) = 16 f ( π ) = 16 f (10) = 16

Piecewise Functions

A piecewise function is a function with two or more formulas for computing y . The formula to use depends on where x is. There will be an interval for x written next to each formula for y .

Introduction to Functions Examples

In this example, there are three formulas for y: y = x − 1, y = 2 x , and y = x ² , and three intervals for x: x ≤ −2, −2 < x < 2, and x ≥ 2. When evaluating this function, we need to decide to which interval x belongs. Then we will use the corresponding formula for y.

Examples

Find f (5), f (−3), and f (0) for the function above.

For f (5), does x = 5 belong to x ≤ −2, −2 < x < 2, or x ≥ 2? Because 5 ≥ 2, we will use y = x ², the formula written next to x ≥ 2.

f (5) = 5 ² = 25

For f (−3), does x = −3 belong to x ≤ −2, −2 < x < 2, or x ≥ 2? Because −3 ≤ −2, we will use y = x − 1, the formula written next to x ≤ −2.

f (−3) = −3 − 1 = −4

For f (0), does x = 0 belong to x ≤ −2, −2 < x < 2, or x ≥ 2? Because −2 < 0 < 2, we will use y = 2 x , the formula written next to −2 < x < 2.

f (0) = 2(0) = 0

Find f (3), f (1), and f (−4) for

Introduction to Functions Examples

f (3) = 5	because 3 > 1
f (1) = −1	because 1 ≤ 1
f (−4) = −(−4) = 4	because − 4 ≤ 1

Piecewise functions come up in daily life. For example, suppose a company pays the regular hourly wage for someone who works up to eight hours but time and a half for someone who works more than eight hours but no more than ten hours and double time for more than ten hours. Then a worker whose regular hourly pay is $10 has the daily pay function below.

Introduction to Functions Examples

Below is an example of a piecewise function taken from an Internal Revenue Service (IRS) publication. The y -value is the amount of personal income tax for a single person. The x -value is the amount of taxable income.

Introduction to Functions Examples

A single person whose taxable income was $30,120 would pay $4341. (Source: 2003, 1040 Forms and Instructions)

Introduction to Functions Practice Problems

Practice

Find f (−1) and f (0) for f ( x ) = 3 x² + 2 x − 1.
Evaluate at x = −3, x = 1, and .
Evaluate g at x = 6, x = 8, and x = 10.
Find f (5), f (3), f (2), f (0), and f (−1).
The function below gives the personal income tax for a single person for the 2003 year. If a single person had a taxable income of $63,575, what is her tax?

Solutions

f (−1) = 3(−1) ² + 2(−1) − 1 = 3 − 2 − 1 = 0 f (0) = 3(0) ² + 2(0) − = −1
The tax is $12,704 because 63,550 ≤ 63,575 < 63,600.

Practice problems for this concept can be found at: Introduction to Functions Practice Test.

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