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Functions &
Function Notation

Master the language of functions: how to read, evaluate, and interpret \( f(x) \) notation for the GRE Quantitative section.

GRE Quantitative

What Is a Function?

A function is a rule that assigns exactly one output to each input. Think of it as a machine: you feed in a number, and it produces precisely one result.

Input
The value you substitute in, often called \( x \) or the argument.
Output
The result the function produces, written as \( f(x) \) or \( y \).

The Function Machine

f rule / formula
\( x \)
 input
\( f(x) \)
 output
Example: \( f(x) = x + 5 \)
 Input 3 → Output 8

Why Functions Matter on the GRE

The GRE Quantitative section frequently tests your ability to evaluate functions, compose functions, and interpret function behavior. These questions reward precision with notation.

Evaluation
Find \( f(a) \) by substituting \( a \) into the rule.
Composition
Apply one function inside another: \( f(g(x)) \).
Interpretation
Read graphs and tables to find inputs or outputs.

Domain and Range

Every function has two key sets. The domain is the set of all valid inputs, and the range is the set of all possible outputs.

Domain Range 1 2 3 4 7 12 f
\( f(x) = x^2 + 3 \)
 1→4, 2→7, 3→12

Core Rule: Evaluating \( f(x) \)

1
Identify the rule. Write down the formula for \( f(x) \).
2
Substitute. Replace every \( x \) with the given input value.
3
Simplify. Follow the order of operations (PEMDAS) carefully.

Composition of Functions

Function composition chains two functions together. The output of the inner function becomes the input of the outer function.

\( x \)
 g
\( g(x) \)
 f
\( f(g(x)) \)
\( (f \circ g)(x) = f(g(x)) \)

\( f(x) \) vs. \( f \cdot x \)

One of the most common sources of confusion on the GRE is mistaking function notation for multiplication. They look similar but mean entirely different things.

\( f(x) \) \( f \cdot x \)
Meaning "Apply function \( f \) to input \( x \)" "Multiply variable \( f \) by \( x \)"
Parentheses Denote function application Would be written \( (f)(x) \) or \( fx \)
Example \( f(3) = 3^2 + 1 = 10 \) \( f \cdot 3 = 3f \)
Context \( f \) is a rule or formula \( f \) is a number or variable

How to Apply Function Notation

1
Read the definition. Identify what \( f(x) \) equals. Write it down clearly.
2
Identify the input. What are you substituting? It could be a number, an expression, or another function.
3
Use parentheses during substitution. Write \( f(\text{input}) \) with the input wrapped in parentheses to avoid sign errors.
4
Simplify step by step. Expand, combine like terms, and reduce. Don't skip steps.
5
Check your answer. Verify by plugging the result back into the context of the problem.

Example 1: Basic Evaluation

1
Substitute \( x = -2 \): \( f(-2) = 2(-2)^2 - 5(-2) + 3 \)
2
Evaluate powers: \( (-2)^2 = 4 \), so \( 2(4) - 5(-2) + 3 \)
3
Multiply: \( 8 + 10 + 3 \)
4
Simplify: \( f(-2) = \) 21

Example 2: Substituting Expressions

1
Replace \( x \) with \( (a+3) \): \( f(a+3) = (a+3)^2 + 1 \)
2
Expand: \( (a+3)^2 = a^2 + 6a + 9 \)
3
Add the constant: \( a^2 + 6a + 9 + 1 \)
4
Result: \( f(a+3) = \) \( a^2 + 6a + 10 \)

Example 3: Composition

1
Work inside out. First evaluate \( g(3) \).
2
Compute \( g(3) \): \( g(3) = 3^2 = \) 9
3
Now evaluate \( f(9) \): \( f(9) = 2(9) + 1 = 18 + 1 = \) 19

Reading Graphs of Functions

On the GRE, you may need to read a graph to find \( f(a) \) or solve \( f(x) = k \). The graph of \( y = f(x) \) plots every input-output pair as a point.

\( x \)
\( y \)
 0 1 2 3 4 0 2 4 (2, 1)
\( f(x) \)

The Vertical Line Test

Not every graph represents a function. A graph is a function if and only if every vertical line crosses it at most once.

✓ Function ✕ Not a Function

Common Mistakes to Avoid

✕ Forgetting Parentheses
For \( f(x) = x^2 \), a common error is writing \( f(-2) = -2^2 = -4 \). The correct way is \( f(-2) = (-2)^2 = 4 \).
✕ Wrong Composition Order
Computing \( g(f(x)) \) when the problem asks for \( f(g(x)) \). Read carefully and work inside out.
✕ Confusing \( f(x+h) \)
Thinking \( f(x+h) = f(x) + f(h) \). Functions are not generally additive: substitute the entire expression.

Summary of Key Concepts

Concept Notation Meaning
Evaluate \( f(a) \) Substitute \( a \) for \( x \)
Compose \( f(g(x)) \) Apply \( g \) first, then \( f \)
Domain Set of inputs All valid \( x \) values
Range Set of outputs All possible \( f(x) \) values
Vertical Line Test Graph check Each \( x \) maps to one \( y \)

Question 1 of 5

If \( f(x) = 3x - 7 \), what is the value of \( f(5) \)?

Question 2 of 5

If \( g(x) = x^2 + 2x \), what is \( g(-3) \)?

Question 3 of 5

If \( f(x) = 2x + 1 \) and \( g(x) = x^2 \), what is \( f(g(2)) \)?

Question 4 of 5

If \( h(x) = \frac{x+1}{x-3} \), which value of \( x \) is NOT in the domain of \( h \)?

Question 5 of 5

If \( f(x) = x^2 - 4x \) and \( f(a) = 5 \), which of the following could be the value of \( a \)?