Functions: Introduction, Notation & Evaluation
Master the foundation of SAT Math — understand what functions are, how to read them, and how to evaluate them like a pro.
Part 1: What Is a Function?
📌 Relations vs. Functions
Relation
A relation is any set of ordered pairs $(x, y)$. It associates inputs with outputs and can be represented as:
- A table of values
- A graph on the coordinate plane
- An equation like $y = x^2 + 1$
- A mapping diagram
Example: $\{(1, 2), (1, 5), (3, 4)\}$ — this IS a relation.
Function ✓
A function is a special relation where each input has exactly one output.
✅ $\{(1, 2), (3, 4), (5, 6)\}$ — Function (each $x$ appears once)
❌ $\{(1, 2), (1, 5), (3, 4)\}$ — NOT a function ($x=1$ maps to both 2 and 5)
✅ $\{(1, 3), (2, 3), (4, 3)\}$ — Function (same $y$ is OK!)
🏭 Think of a Function as a Machine
You put an input ($x$) in → the machine applies a rule → you get exactly one output ($y$) out. If you put the same input in twice, you MUST get the same output both times.
📌 Key Vocabulary
Input
Independent variable
$x$
Output
Dependent variable
$y = f(x)$
Domain
All possible inputs
What you CAN put in
Range
All possible outputs
What you CAN get out
🌟 Why Functions Matter on the SAT
Functions appear in 15–20% of SAT Math questions. They model real-world relationships (cost, distance, population), predict outcomes, and form the foundation of algebra and calculus. Understanding functions is essential for success.
Part 2: Function Notation
📌 Understanding $f(x)$
Function notation uses the format $f(x)$, read as "f of x".
| Symbol | Meaning |
|---|---|
| $f$ | The name of the function (could be $g$, $h$, $p$, etc.) |
| $x$ | The input value |
| $f(x)$ | The output value (same as $y$) |
So $y = 2x + 3$ and $f(x) = 2x + 3$ mean the exact same thing.
🚨 Critical Warning
$f(x)$ does NOT mean $f \times x$
It means "the function $f$ evaluated at $x$." This is the #1 notation mistake students make!
Different Function Names
- $f(x) = 2x + 3$ — function "f"
- $g(x) = x^2 - 1$ — function "g"
- $h(x) = \sqrt{x}$ — function "h"
- $p(t) = 100 \cdot 2^t$ — function "p" with input $t$
What $f(x)$ Tells You
If $f(x) = 2x + 3$, then $f$ represents the rule:
"Double the input and add 3"
- $f(1) = 2(1) + 3 = 5$
- $f(4) = 2(4) + 3 = 11$
- $f(a) = 2a + 3$
💡 Key Insight: $f(x)$ = $y$
Whenever you see $f(x)$, you can think of it as the $y$-value. The equation $y = 2x + 3$ is identical to $f(x) = 2x + 3$. The point $(a, f(a))$ lies on the graph of $f$.