Functions are very much like the “magic boxes” you may have learned about in elementary school. For example:
You put a 2 into the magic box, and a 7 comes out. You put a 3 into the magic box, and a 9 comes out. You put a 4 into the magic box, and an 11 comes out. What is the magic box doing to your number?
There are many possible ways to describe what the magic box is doing to your number. One possibility is as follows: The magic box is doubling your number and adding 3:
| 2(2) + 3 = 7 | 2(3) + 3 = 9 | 2(4) + 3 = 11 |
It’s possible that the magic box is actually doing something different to your number. But assuming that the box is really following the "double and add 3" rule, you can express this result this way: 2x + 3. This whole process can be written in function form as:
f(x) = 2x + 3
In words, you say “f of x equals 2x plus 3.” Here, x is the number you put in the box, f is the magic box itself, and the value of 2x + 3 is the result that comes out of the box. Functions always work the same way every time, so you can think of the function f as actually representing the rule that the magic box is using to transform your number.
The magic box analogy is a helpful way to conceptualize a function as a rule built on an independent variable. The value of a function changes as the value of the independent variable changes. In other words, the value of a function is dependent on the value of the independent variable. Examples of functions include:
| f(x) = 4x2 − 11 | The value of the function f is dependent on the independent variable x. |
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The value of the function g is dependent on the independent variable t. |
Think of functions as consisting of an “input” variable (the number you put into the magic box), and a corresponding “output” value (the number that comes out of the box). The function is simply the rule that turns the “input” variable into the “output” variable.
By the way, the expression f(x) is pronounced “f of x”, not “fx.” It does not mean “f times x.”The letter f does not stand for a variable; rather, it stands for the rule that dictates how the input x changes into the output f(x).
The “domain” of a function indicates the set of possible inputs. The “range” of a function indicates the set of possible outputs. For instance, the function f(x) = x2 can take any input but never produces a negative number. So the domain is all numbers, but the range is f(x) ≥ 0.
This is the most basic type of function problem. Input the numerical value (say, 5) in place of the independent variable x to determine the value of the function:
If f(x) = x2 − 2, what is the value of f(5)?
In this problem, you are given a rule for f(x): square x and subtract 2. Then, you are asked to apply this rule to the number 5. Square 5 and subtract 2 from the result:
f(5) = (5)2 − 2 = 25 − 2 = 23
This type of problem is slightly more complicated. Instead of finding the output value for a numerical input, you must find the output when the input is an algebraic expression:
If f(z) = z2 −
, what is the value of f(w + 6)?
Input the variable expression (w + 6) in place of the independent variable (z) to determine the value of the function:
Compare this equation to the equation for f(z). The expression (w + 6) has taken the place of every z in the original equation. In a sense, you are treating the expression (w + 6) as one thing, as if it were a single letter or variable.
The rest is algebraic simplification:
Imagine putting a number into one magic box, and then putting the output directly into another magic box. This is the situation you have with compound functions:
The expression f(g(3)), pronounced “f of g of 3,” looks ugly, but the key to solving compound function problems is to work from the inside out. In this case, start with g(3). Notice that you put the number into g, not into f, which may seem backward at first:
g(3) = 4(3) − 3 = 12 − 3 = 9
Use the result from the inner function g as the new input variable for the outer function f:
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The final result is 732. |
Note that changing the order of the compound functions changes the answer:
Again, work from the inside out. This time, start with f(3), which is now the inner function:
Use the result from the inner function f as the new input variable for the outer function g:
Thus,
In general, f(g(x)) and g(f(x)) are not the same rule overall and will often lead to different outcomes. As an analogy, think of “putting on socks” and “putting on shoes” as two functions: the order in which you perform these steps obviously matters!
You may be asked to find a value of x for which f(g(x)) = g(f(x)). In that case, use variable substitution, working as always from the inside out:
If f(x) = x3 + 1, and g(x) = 2x, for what value of x does f(g(x)) = g(f(x))?
Simply evaluate as you did in the problems above, using x instead of an input value:
On the GRE, you may be given a function with an unknown constant. You will also be given the value of the function for a specific number. You can combine these pieces of information to find the complete function rule:
If f(x) = ax2 − x, and f(4) = 28, what is f(−2)?
Solve these problems in three steps. First, use the value of the input variable and the corresponding output value of the function to solve for the unknown constant:
Then, rewrite the function, replacing the constant with its numerical value:
f(x) = ax2 − x = 2x2 − x
Finally, solve the function for the new input variable:
f(−2) = 2(−2)2 −(−2) = 8 + 2 = 10
A function can be visualized by graphing it in the coordinate plane. The input variable is considered the domain of the function, or the x-coordinate. The corresponding output is considered the range of the function, or the y-coordinate.
What is the graph of the function f(x) = −2x2 + 1?
| INPUT | OUTPUT | (x, y) |
| −3 | −2(−3)2 + 1 = −17 |
(−3, −17) |
| −2 | −2(−2)2 + 1 = −7 |
(−2, −7) |
| −1 | −2(−1)2 + 1 = −1 |
(−1, −1) |
| 0 | −2(0)2 + 1 = 1 |
(0, 1) |
| 1 | −2(1)2 + 1 = −1 |
(1, −1) |
| 2 | −2(2)2 + 1 = −7 |
(2, −7) |
| 3 | −2(3)2 + 1 = −17 |
(3, −17) |
Create an input-output table by evaluating the function for several input values.
Then, plot points to see the shape of the graph: 
If
, what is f(−1)?
If t(u) = au2 − 3u + 1 and t(3) = 37, what is a?
If f(x) = 3x −
and g(x) = x2, what is g(f(4))?
If
, what is
?
