Functions are generally presented in the form of equations on the Praxis Core. A function may look scary with the f(x) notation at the beginning of the equation, but you have nothing to worry about. If you can solve basic equations, you can solve functions.
Identifying functions
First, you need to understand some other basic terminology. To start with, know that a set of ordered pairs is a relation. For example, {(3, 5), (7, 10), (8, –1)} is a relation. It is a set of three ordered pairs. Relations can be represented in other ways. A table is a means of representing ordered pairs by listing x-coordinates next to the y-coordinates with which they are paired.
| x | y |
|---|---|
| –7 | –2 |
| –1 | 4 |
| 2 | 3 |
| 5 | 0 |
The table represents the ordered pairs (–7, –2), (–1, 4), (2, 3), and (5, 0).
Relations can also be represented by points on the coordinate plane and by graphs of equations. The graph of an equation represents an infinite number of ordered pairs.
The set of x values in a relation is the domain, and the set of y values is the range of a relation. Variables other than x and y can be represented by a relation. However, universally, the domain of a relation is the set of the ordered pairs’ first variable values, and the range is the set of second variable values.
Now that you are familiar with the terms relation, domain, and range, you’re ready to see the bigger picture of functions. A function is a relation in which each number in the domain is paired with only one number in the range.
Generally, since the first variable of the ordered pairs in a function tends to be x, a function involves x but no repeat of an x value. Each domain value is paired with just one range value, so a value of x never repeats, unless the same range value repeats with it, which is rare.
However, a range value can repeat in a function without the same domain value repeating with it.
The requirement for a function is that no number in the domain is paired with more than one number in the range, not that no number in the range is paired with more than one number in the domain.
The relation {(1, 2), (1, 3), (1, 4)} is not a function because 1 is paired with three different range values, but the relation {(1, 5), (2, 5), (3, 5)} is a function. The fact that 5 is paired with three different domain values does not matter. 5 is a range value
In a function in which the numbers represent x and y, for every x value, only one y value exists.
Which of the following relations is NOT a function?
(A) {(4, 8), (5, –1), (7, 6), (10, 4)}
(B) {(–2, 7), (–1, 2), (5, –4), (5, –4), (19, 0), (22, 7)}
(C) {(0, 1), (1, 2), (2, 3), (3, 4), (4, 5)}
(D) {(–5, 10), (0, 10), (5, 10), (10, 10)}
(E) {(2, 4), (4, 6), (6, 7), (2, 9), (7, 1)}
The correct answer is Choice (E). The domain number 2 is repeated and paired with both 4 and 9. Thus, 2 is paired with more than one range number. That means that the relation is not a function. Choice (A) is incorrect because no domain number is paired with more than one range number.
Choice (B) is incorrect, because although the domain number 5 is repeated, 5 is only paired with –4. Choice (C) is incorrect because, although some numbers are used more than once, no domain number is paired with more than one range number. Choice (D) is incorrect because, although 10 is a range number three times, no domain number is paired with more than one range number.
Working with functions
Functions in the forms of equations often involve f(x), or another letter followed by x, set equal to an expression that contains x. f(x) is pronounced “f of x.”
Consider the equation f(x) = x + 5. Any value that you put in for x will result in just one value of f(x). A value that is to stand in for x will be represented in the parentheses next to f to show that the value takes the place of x.
For the function f(x) = x + 5, you can determine the value of f(12) by putting 12 in for x in x + 5. The result is 12 + 5, or 17. 12 takes the place of x in f(x), so 12 takes the place of x in x + 5. Understanding that principle is the key.
Since the letter next to the parentheses is f, the name of the function is f. Letters other than f are often used in function equations. For example, g(x), h(x), and p(x) are commonly used.
If g(x) = x2 + 3, what is the value of g(5)?
(A) 5
(B) 8
(C) 28
(D) 25
(E) 3
The correct answer is Choice (C). Because 5 takes the place of x in g(x), 5 takes the place of x in x2 + 3. Therefore, g(5) = 52 + 3, which is 25 + 3, or 28.
Choice (A) is just the number that replaces x. Choice (B) is the value of 5 + 3 instead of 52 + 3. Choice (D) is merely the value of 52. Choice (E) is just the number that is added to x2 in the function.