Functions: An introduction
Mathematics is known for its ability to convey a great deal of information with the use of the minimum number of symbols. While this may be initially confusing (if not frustrating) for the learner, the notation of mathematics is a universal language. In this chapter, you will learn about function notation.
Relations and inverses
One of the major concepts used in mathematics is relations. A relation is any set of ordered pairs. The set of all first elements (the input values) is called the domain, while the set of second elements (the output values) is called the range. Relations are traditionally named with a capital letter. For example, given the relation
A = {(2, 3), (−1, 5), (4, −3), (2, 0), (−9, 1)}
the domain of A (written DA) is {−9, −1, 2, 4}. The domain was written in increasing order for the convenience of reading, but this is not required. The element 2, which appears as the input for two different ordered pairs, needs to be written only one time in the domain. The range of A (written RA) is {−3, 0, 1, 3, 5}.
The inverse of a relation is found by interchanging the input and output values. For example, the inverse of A (written A−1) is
A−1 = {(3, 2), (5, −1), (−3, 4), (0, 2), (1, −9)}
Do you see that the domain of the inverse of A is the same set as the range of A, and that the range of the inverse of A is the same as the domain of A? This is very important.
Given the relationships:
1. Find the domain of A.
2. Find the range of A.
3. Find the domain of B.
4. Find the range of B.
5. Find the domain of C.
6. Find the range of C.
7. Find A−1.
8. Find B−1.
9. Find C−1.
Functions
Functions are a special case of a relation. By definition, a function is a relation in which each element of the domain (the input value) has a unique element in the range (the output value). In other words, for each input value there can be only one output value. Looking at the relations for A and A−1 in the previous section, you can see that A is not a function because the input value of 2 is associated with the output values 3 and 0. The relation A−1 is a function because each input value is paired with a unique output value. (Don’t be confused that the number 2 is used as an output value for two different input values. The definition of a function does not place any stipulations on the output values.)
Given the relationships:
1. Which of the relations A, B, and C are functions?
2. Which of the relations A−1, B−1, and C−1 are functions?
3. A relation is defined by the sets {(students in your math class), (telephone numbers at which they can be reached)}. That is, the input is the set of students in your math class and the output is the set of telephone numbers at which they can be reached. Must this relationship be a function? Explain.
4. Is the inverse of the relation in question 3 a function? Explain.
5. A relation is defined by the sets {(students in your math class), (the student’s Social Security number)}. Must this relationship be a function? Explain.
6. Is the inverse of the relation in question 5 a function? Explain.
7. A relation is defined by the sets {(students in your math class), (the student’s birthday)}. Must this relationship be a function? Explain.
8. Is the inverse of the relation in question 7 a function? Explain.
Function notation
Function notation is a very efficient way to represent multiple functions simultaneously while also indicating domain variables. Let’s examine the function f(x) = 5x + 3. This reads as “f of x equals 5x + 3.” The name of this function is f, the independent variable (the input variable) is x, and the output values are computed based on the rule 5x + 3. In the past, you would have most likely just written y = 5x + 3 and thought nothing of it. Given that, be patient as you work through this section.
What is the value of the output of f when the input is 4? In function notation, this would be written as f(4) = 5(4) + 3 = 23. Do you see that the x in the name of the function is replaced with a 4—the desired input value—and that the x in the rule of this function is also replaced with a 4? The point (4, 23) is a point on the graph of this function.
Consequently, you should think of the phrase y = whenever you read f(x). That is, if the function reads f(x) = 5x + 3 you should think y = f(x) = 5x + 3 so that you will associate the output of the function with the y-coordinate on the graph. f(−2) = 5(−2) + 3 = −7 indicates that when −2 is the input, −7 is the output, and the ordered pair (−2, −7) is a point on the graph of this function.
Consider a different function, g(x) = −3x2 + 2x + 5. g(2) = −3(2)2 + 2(2) + 5 = −12 + 4 + 5 = −3. The point (2, −3) is on the graph of the parabola defined by g(x). If h(t) = −16t2 + 128t + 10, h(3) = −16(3)2 + 128(3) + 10 = −144 + 384 + 10 = 250. In essence, function notation is a substitution-guided process. Whatever you substitute within the parentheses on the left-hand side of the equation is also substituted for the variable on the right-hand side of the equation.
Given f(x) = −3x + 8, find
1. f(−4)
2. f(5)
3. f(n + 2)
4. g(5)
5. g(−2)
6. g(t − 1)
7. p(5)
8. p(−1)
9. p(r − 2)
Arithmetic of functions
Arithmetic can be performed on functions. For example, let g(x) = 7x − 2 and 
To calculate g(2) + p(2), you first evaluate each of the functions [g(2) = 12 and p(2) = 8] and then add the results: g(2) + p(2) = 20. g(3) − p(1) shows that the input values do not have to be the same to do arithmetic. g(3) = 19 and p(1) = −5, so g(3) − p(1) = 24.
What does p(g(2)) equal? A better question to answer first is what does p(g(2)) mean? Since g(2) is inside the parentheses for the function p, you are being told to make that substitution for x in the rule for p. It will be more efficient (and involve less writing) if you first determine that g(2) =12 and evaluate p(12). 
Therefore, 
Evaluating a function with another function is called composition of functions. While 
, g(p(2)) = g(8) = 7(8) − 2 = 54. This illustrates that you must evaluate a composition from the inside to the outside.
As you know, there are two computational areas that will not result in a real number answer; thus, you do not (1) divide by zero, or (2) take the square root (or an even root) of a negative number. These rules are useful when trying to determine the domains of functions.
Finding the range of a function is more challenging. This topic will be brought up throughout this book as particular types of functions are studied.
Given f(x) = 2x2− 3x and 
answer questions 1–6.
1. f(2) + g(5)
2. g(f(2))
3. f(1) × g(0)
4. 
5. f(g(21))
6. g(33) − f(−2)
Find the domain for each of the following functions.
7. 
8. 
9. 
10. 
Transformation of functions
The graphs of y = x2, y = x2 + 3, y = x2 − 6, y = (x −4)2, y = (x + 5)2, and y = (x + 1)2 − 2 are all parabolas. The difference among them is their location on the plane. Understanding the behavior of the base function, y = x2, and the transformation that moves this function to a new location gives a great deal of information about the entire family of parabolas. Examine the following graphs.
The graph of y = x2 + k is a vertical translation of the graph of y = x2. If k > 0, the graph moves up, and if k < 0, the graph moves down. The graph of y = (x − h)2 is a horizontal translation of the graph of y = x2. The graph moves to the right when h > 0 and to the left when h < 0.
The transformation of y = x2 to get the graph of y = (x + 1)2 − 2 is a combination of the two. The graph of the parabola moves to the left 1 unit and down 2 units.
The graph of y = ax2 is a stretch from the x-axis. It is important that you do not confuse the dilation from the origin that you studied in geometry (in which both the x- and y-coordinates are multiplied by the stretch factor) with a dilation from the x-axis (in which only the y-coordinate is multiplied by the stretch factor). If 0 < a < 1, the graph moves closer to the x-axis, while if a > 1, the graph moves further from the x-axis. If a < 0, the graph is reflected over the x-axis.
Describe the transformation of each of the base functions y = x2, 
or y = |x|, whichever is appropriate.
1. f(x) = 3(x + 2)2 − 1
2. g(x) = |x − 1| − 3
3. 
4. p(x) = −2x2 + 3
5. 
Inverse of a function
To find the inverse of a function, the same notion of interchanging the x- and y-coordinates are applied. For example, to find the inverse of f(x) = 5x + 3, think about the function as y = 5x + 3. Switch the x and y: x = 5y + 3. Because as functions are written in the form y = rather than x =, solve the equation for y. Subtract 3 to get x − 3 = 5y and then divide by 5 to get 
If f(x) = 5x + 3 then 
For each function given, find the inverse.
1. Given f(x) = 3x − 5, find f−1(x).
2. Given g(x) = 5 − 8x, find g−1(x).
3. 
4. 
Graphical representation of functions
Sets of ordered pairs are useful for clarifying the concepts of relation, function, inverse, domain, and range, but as you know, most of mathematics is done with formulas and graphs. By definition, a function is a relation in which no input value has multiple output values associated with it. What does that look like on a graph? The definition would indicate that it would not be possible to draw a vertical line anywhere on the graph and have it hit more than one of the plotted points at any one time. (If the vertical line does not hit any of the points, that is fine. The requirement is that the vertical line cannot hit more than one point at a time.)
At first, it is not as easy to determine if a relation represents a function when only given an equation. With experience, you will be able to tell which equations will probably not represent functions, and which are likely to. For example, you most likely recognize that the equation x2 + y2 = 36 represents a circle with its center at the origin and a radius of 6. This is not a function. You also know that the equation y = 3x2 is a parabola that opens up and has its vertex at the origin. This is a function. Do you know what the graphs of x = 3y2 or xy3 − x3y = 12 look like? Neither is a function, and this can be shown by picking a value for x (e.g., x = 1) and noting that there is more than one value of y associated with it. Fortunately, you will not encounter these equations while studying Algebra II.
Finding the inverse from a graph is not easy. Determining whether the graph of a relation is a function is not as difficult. Recall that the vertical line test is used to determine if a graph represents a function. If the inverse of the relation defined by the graph is to be a function, then none of the y-coordinates can be repeated (if they were, then the graph of the inverse would fail the vertical line test). If the y-coordinates cannot be repeated, then the graph would have to pass a horizontal line test. To recap this important information:
If a relation passes the vertical line test, the relation is a function.
If a relation passes the horizontal line test, the inverse of the relation is a function.
Relations that pass both the vertical and horizontal line tests are called one-to-one functions.
Use these graphs to answer questions 1 and 2.
1. Which of the relations defined by the graphs given represents a function?
2. Which of the relations defined by the graphs given will have inverses that are functions?