Function Notation
Function notation is a way to name a function and show its input. In f(x) = 2x + 3, f is the name of the function, x is the input, and f(x) is the output. To evaluate f(4), replace x with 4: f(4) = 2(4) + 3 = 11.
Before this lesson: You should know what a function is from relations and functions, and how to evaluate algebraic expressions.
What Does f(x) Mean?
You read f(x) as “f of x.” It does not mean f times x. It means “the output of the function f when the input is x.”
Think of a function as a machine. You drop in an input. The machine follows its rule. Out comes one output. The notation f(4) asks, “What comes out when 4 goes in?”
Function notation is just a label for y. The equation y = 2x + 3 and the function f(x) = 2x + 3 make the same line. The fact f(4) = 11 means the point (4, 11) is on the graph.
Why use it? The name tells you which function you mean, and the input is right there. If a problem has two functions, you can call them f and g and never mix them up.
How to Evaluate a Function
To evaluate a function means to find the output for a given input.
- Write the rule. Replace every x with the input. Put the input in parentheses.
- Simplify with the order of operations.
- Write the answer as f(\text{input}) = \text{output}.
Some problems go the other way. They give the output and ask for the input. For “Find x when f(x) = 19,” set the rule equal to 19 and solve the equation.
| The question | What it asks | What you do |
|---|---|---|
| Find f(3) | The output when the input is 3 | Replace x with 3 and simplify |
| Solve f(x) = 3 | The input that gives an output of 3 | Set the rule equal to 3 and solve |
Function Notation Examples
Example 1: Evaluate a linear function
Let f(x) = 3x - 5. Find f(4).
\begin{aligned}
f(4) &= 3(4) - 5 \\
&= 12 - 5 \\
&= 7
\end{aligned}Answer: f(4) = 7.
Example 2: A negative input and a square
Let g(x) = x^2 - 2x + 1. Find g(-3).
Replace both x‘s with (-3). The parentheses keep the sign with the number.
\begin{aligned}
g(-3) &= (-3)^2 - 2(-3) + 1 \\
&= 9 + 6 + 1 \\
&= 16
\end{aligned}Answer: g(-3) = 16.
Example 3: Find the input from the output
Let h(x) = 4x + 7. Find x when h(x) = 19.
\begin{aligned}
4x + 7 &= 19 \\
4x &= 12 \\
x &= 3
\end{aligned}Check: h(3) = 4(3) + 7 = 19. It works.
Answer: x = 3.
Example 4: Read a function from a graph
The graph shows f(x) = 2x - 1. Find f(2). Then find x when f(x) = -3.
Find f(2): start at 2 on the x-axis. Go up to the line. You land at height 3. So f(2) = 3.
Find x when f(x) = −3: start at -3 on the y-axis. Go across to the line. It meets the line at x = -1. Check: f(-1) = 2(-1) - 1 = -3.
Answer: f(2) = 3, and f(x) = -3 when x = -1.
Example 5: Word problem
A plumber charges a $60 trip fee plus $45 per hour. The cost for h hours is C(h) = 45h + 60 dollars. Find C(3) and explain what it means. Then find how many hours cost $285.
C(3) = 45(3) + 60 = 135 + 60 = 195
So C(3) = 195. A 3-hour job costs $195.
\begin{aligned}
45h + 60 &= 285 \\
45h &= 225 \\
h &= 5
\end{aligned}Answer: A 3-hour job costs $195. A bill of $285 is for 5 hours of work.
Common Mistakes
| Mistake | Example of the error | Fix |
|---|---|---|
| Reading f(x) as multiplication | Writing f(2) = 2f | f(2) means the output when the input is 2. |
| Leaving out parentheses with a negative input | For x^2 at x = -3, writing -3^2 = -9 | Write (-3)^2 = 9. |
| Mixing up f(3) and f(x) = 3 | Plugging in 3 when asked to solve f(x) = 3 | f(x) = 3 gives the output. Set the rule equal to 3 and solve. |
| Replacing only one x | For g(x) = x^2 - 2x, finding g(4) = 16 - 2x | Replace every x with the input. |
Function Notation in Tables and Other Letters
A function can be given as a table. To find f(1), find 1 in the x row and read the output below it.
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| f(x) | 4 | 1 | -2 | -5 |
Here f(1) = -2. To solve f(x) = 4, find 4 in the bottom row. Its input is -1, so x = -1.
Functions don’t have to use f and x. Word problems often use letters that match the meaning, like C(h) for cost after h hours or d(t) for distance after t seconds. The steps are the same.
You will use function notation for lines in slope-intercept form: f(x) = mx + b is the same line as y = mx + b.
Practice Problems
Use f(x) = 5x + 2, g(x) = x^2 - 4 and h(x) = -2x + 9 for problems 1–7.
- Find f(3).
- Find f(-2).
- Find g(5).
- Find g(-1).
- Find h(0).
- Find x when f(x) = 27.
- Find x when h(x) = 15.
- Let p(x) = \frac{x}{2} + 3. Find p(-6).
- Use the table in the section above. Find f(1), and find x when f(x) = 4.
- A phone plan costs C(m) = 0.10m + 15 dollars for m minutes. Find C(200). How many minutes cost $40?
Show answers
- f(3) = 5(3) + 2 = 17
- f(-2) = 5(-2) + 2 = -10 + 2 = -8
- g(5) = 5^2 - 4 = 25 - 4 = 21
- g(-1) = (-1)^2 - 4 = 1 - 4 = -3
- h(0) = -2(0) + 9 = 9
- 5x + 2 = 27, so 5x = 25 and x = 5.
- -2x + 9 = 15, so -2x = 6 and x = -3.
- p(-6) = \frac{-6}{2} + 3 = -3 + 3 = 0
- f(1) = -2. The output 4 has input -1, so x = -1.
- C(200) = 0.10(200) + 15 = 35, so 200 minutes cost $35. For $40: 0.10m + 15 = 40, so 0.10m = 25 and m = 250 minutes.
Function Notation FAQ
What does f(x) mean?
f(x) means the output of the function f when the input is x. You read it as “f of x.”
How do you evaluate a function?
Replace every x in the rule with the input, using parentheses. Then simplify with the order of operations.
Is f(x) the same as y?
Yes, for graphing. The graph of f(x) = 2x + 3 is the same as the graph of y = 2x + 3. Function notation also shows the input.
Does f(x) mean f times x?
No. The parentheses here hold the input. They do not mean multiplication.
Summary
- f(x) is read “f of x.” It is the output when the input is x.
- To find f(a), replace every x with a in parentheses, then simplify.
- To solve f(x) = c, set the rule equal to c and solve for x.
- f(a) = b means the point (a, b) is on the graph.
- Functions can use any letters, like C(h) for cost after h hours.
Previous lesson: Relations and Functions
Next lesson: Domain and Range
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