SAT Math Mastery: Chapter 2 – Functions & Function Notation | satmath800.com

SAT Math Mastery • Chapter 2

Functions & Function Notation

Same rule, every time. Read the machine, not just the numbers.

What You Will Master in This Chapter

Read and use function notation f(x), and evaluate functions from equations, tables, and graphs
Determine the domain and range of a function, including restrictions from denominators and radicals
Interpret function values and rates of change in real-world contexts
Read key features directly off a graph: intercepts, increasing/decreasing intervals, and max/min points
Predict how shifts, reflections, and stretches change a function’s graph and equation
Evaluate and build composite functions, working carefully from the inside out

SAT SNAPSHOT: Function notation sits at the hinge between the Algebra and Advanced Math domains — the same f(x) fluency you build here gets reused constantly once linear equations give way to quadratics and exponentials in later chapters. Expect function questions in every format the SAT uses: algebraic, tabular, graphical, and real-world context.

2.1 Foundations — What Is a Function? 2.2 Evaluating Functions 2.3 Domain and Range 2.4 Functions in Context — Word Problems 2.5 Graphs of Functions — Key Features 2.6 Transformations of Functions 2.7 Composite Functions 2.8 Chapter Formula Recap 2.9 Mixed Practice Set Answer Key & Explanations

2.1 Foundations — What Is a Function?

A function is a rule that assigns exactly one output to each input. For every value you put in, the function hands back exactly one value — never two, never none. This single idea is the foundation for everything else in this chapter.

FUNCTION NOTATION:
f(x) is read “f of x.” It means: the output the function f produces when the input is x.

The letter f is just a name — you will see functions called g, h, P, C, N, or almost any letter. The parentheses do not mean multiplication. This trips up more students than any other notation on the entire test.

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SAT PATTERN ALERT f(x) is NOT f times x. If you see f(3), that means “substitute 3 for x and evaluate” — it has nothing to do with multiplying f by 3.

Is It Actually a Function? The Vertical Line Test

For a graph, there’s a fast visual check: if any vertical line crosses the graph more than once, it is not a function (that input would have two different outputs, which breaks the rule).

Figure 1. A graph represents a function only if every vertical line crosses it at most once.

THE SAME IDEA IN A TABLE

A table or a set of ordered pairs represents a function as long as no x-value repeats with two different y-values. The pairs (2, 5) and (2, 9) together would break the rule — an input of 2 can’t produce two different outputs.

2.2 Evaluating Functions

Evaluating a function means finding its output for a specific input. Whether the function is given as an equation, a table, or a graph, the process is the same: locate the input, then read or calculate the matching output.

HOW TO EVALUATE f(a):

From an equation: substitute a for every x, then simplify.
From a table: find a in the input row/column and read the paired output.
From a graph: find x = a on the graph and read the y-coordinate there.

EXAMPLE 2.1 — Direct Evaluation from an Equation

If f(x) = 4x – 3, what is the value of f(6)?

(A) 15

(B) 18

(D) 24

Substitute 6 for every x in the equation: f(6) = 4(6) – 3

Simplify: f(6) = 24 – 3 = 21

Answer: (C) 21

EXAMPLE 2.2 — Evaluation from a Table

The table below shows several values of the function f. What is the value of f(4) – f(1)?

x	0	1	2	3	4
f(x)	2	5	8	11	14

(A) 3

(B) 6

(D) 12

Read the table directly: f(4) = 14 and f(1) = 5.

Subtract: f(4) – f(1) = 14 – 5 = 9

Answer: (C) 9

EXAMPLE 2.3 — Evaluation from a Graph

The graph of y = f(x) is shown below. What is the value of f(2)?

Figure 2. Read the y-coordinate directly above/below x = 2.

(A) −2

(B) −1

(D) 1

Locate x = 2 on the graph and trace down to the curve.

The curve passes through the point (2, -1) at that location.

Answer: (B) f(2) = -1

EXAMPLE 2.4 — Evaluating with a Negative Input (Watch the Exponent)

If f(x) = 2x² – 5, what is the value of f(-3)?

(A) −23

(B) −11

(D) 13

Substitute −3 for x, keeping it inside parentheses so the negative gets squared too: f(-3) = 2(-3)² – 5

Square first: (-3)² = 9, so: f(-3) = 2(9) – 5

Simplify: f(-3) = 18 – 5 = 13

Answer: (D) 13

EXAMPLE 2.5 — Working Backward — Finding the Input from the Output

The function f is defined by f(x) = 3x – 7. If f(k) = 11, what is the value of k?

(A) 4

(B) 5

(D) 7

Set the expression for f(k) equal to 11: 3k – 7 = 11

Add 7 to both sides: 3k = 18

Divide both sides by 3: k = 6

Answer: (C) k = 6

2.3 Domain and Range

The domain of a function is the complete set of inputs (x-values) the function is allowed to accept. The range is the complete set of outputs (y-values) it can actually produce. Most SAT domain questions boil down to spotting one of two red flags.

TWO DOMAIN RESTRICTION RULES:

Rule 1 — Denominators: A fraction is undefined when its denominator equals 0. Exclude any x-value that makes the denominator 0.
Rule 2 — Even Radicals: A square root (or any even root) of a negative number is not a real number. The expression under the radical must be ≥ 0.

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SAT PATTERN ALERT The SAT frequently combines both rules in one function, or hides the restriction inside a real-world scenario (like time or distance, which can’t be negative). Always ask: what could break this function?

RANGE, BRIEFLY

There’s no single formula for range — you typically find it by reasoning about the function’s behavior (does it have a minimum or maximum? does it grow without bound?) or by reading the extent of a graph from bottom to top.

EXAMPLE 2.6 — Domain — Denominator Equals Zero

For the function f(x) = 5x – 4, which value of x is NOT in the domain?

(A) −5

(B) −4

(D) 5

The function is undefined when the denominator equals 0. Set it equal to 0: x – 4 = 0

Solve: x = 4

Answer: (C) x = 4

EXAMPLE 2.7 — Domain — Even Radical

What is the domain of f(x) = √x + 6?

(A) x ≥ −6

(B) x ≤ −6

(D) x ≤ 6

The expression under the radical must be greater than or equal to 0: x + 6 ≥ 0

Subtract 6 from both sides: x ≥ -6

Answer: (A) x ≥ -6

EXAMPLE 2.8 — Domain — A Slightly Busier Denominator

For the function f(x) = x + 32x – 10, which value of x must be excluded from the domain?

(A) −3

(B) 0

(D) 5

Set the denominator equal to 0: 2x – 10 = 0

Add 10 to both sides, then divide by 2: 2x = 10 → x = 5

Answer: (D) x = 5

EXAMPLE 2.9 — Domain in a Real-World Context

A ball is launched into the air, and its height in feet, t seconds after launch, is modeled by h(t) = -16t² + 40t + 5, until the ball lands. Which of the following is the most appropriate restriction on the domain of h?

(A) t ≥ 0

(B) t ≤ 0

(D) t > 5

Time cannot be negative in this real-world scenario — the model only makes sense from the moment of launch onward.

So the domain should be restricted to non-negative values of t.

Answer: (A) t ≥ 0

EXAMPLE 2.10 — Range from a Graph Description

The graph of y = f(x) has a single minimum point at (1, −2), opens upward from that point, and extends without bound in both directions along the x-axis. What is the range of f?

(A) y ≥ −2

(B) y ≤ −2

(D) all real numbers

Since (1, −2) is the lowest point on the graph and the graph extends upward from there without bound, the smallest output is −2 and every larger output is possible.

Answer: (A) y ≥ -2

2.4 Functions in Context — Word Problems

Real-world function questions dress up the same evaluation and solving skills from Sections 2.2 in a story problem. The extra challenge is entirely about reading comprehension: matching each number in the function to what it represents in the scenario.

ATTACKING INTERPRETATION QUESTIONS:

Identify the input: what real-world quantity does x represent, and what are its units?
Identify the output: what real-world quantity does f(x) represent, and what are its units?
Match direction: does the answer choice describe input causing output, or does it accidentally reverse the two?

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SAT PATTERN ALERT The four answer choices on an interpretation question often describe the same numbers in different, almost-identical sentences. The SAT is testing whether you can tell “the cost after 5 hours” apart from “the number of hours it takes to reach a cost of 5.” Read every choice slowly — this is not a place to skim.

EXAMPLE 2.11 — Direct Model Evaluation

The function C(x) = 45 + 0.25x gives the cost, in dollars, to rent a bike for x miles ridden. What is the cost to ride the bike 60 miles?

(A) $45.00

(B) $52.50

(D) $75.00

Substitute 60 for x: C(60) = 45 + 0.25(60)

Simplify: C(60) = 45 + 15 = 60

Answer: (C) $60.00

EXAMPLE 2.12 — Interpreting a Function Statement

The function P(t) = 850 – 12t models the number of pages Maria has left to read in a book, t days after she began reading, for 0 ≤ t ≤ 60. What is the best interpretation of the statement P(10) = 730?

(A) 10 days after Maria began reading, she has 730 pages left to read.

(B) 10 days after Maria began reading, she has read 730 pages.

(D) 730 days after Maria began reading, she has 10 pages left to read.

In this model, t is the number of days (the input) and P(t) is the number of pages left to read (the output).

So P(10) = 730 means: when the input is 10 days, the output — pages remaining — is 730.

Choices (B) and (C) both swap “left to read” for “read,” and choice (D) swaps the input and output.

Answer: (A) 10 days after Maria began reading, she has 730 pages left to read.

EXAMPLE 2.13 — Interpreting a Rate of Change

The number of books in a library’s collection is modeled by N(t) = 15,000 + 500t, where t is the number of years since the library opened. What does the number 500 represent in this context?

(A) the number of books in the collection when the library opened

(B) the number of years the library has been open

(D) the total number of books in the collection after t years

In slope-intercept style, 15,000 is the starting value (at t = 0) and 500 is multiplied by t — the rate at which the collection changes per year.

Answer: (C) the number of books added to the collection each year

EXAMPLE 2.14 — Solving for the Input in Context

Using the same function from item 2.13, N(t) = 15,000 + 500t, after how many years will the collection reach 21,000 books?

(A) 10

(B) 11

(D) 13

Set N(t) equal to 21,000: 15,000 + 500t = 21,000

Subtract 15,000 from both sides: 500t = 6,000

Divide both sides by 500: t = 12

Answer: (C) t = 12

EXAMPLE 2.15 — Context with an Inequality

A company’s profit, in dollars, from selling x units of a product is modeled by f(x) = 80x – 1,200. What is the minimum whole number of units the company must sell for its profit to be positive?

(A) 14

(B) 15

(D) 17

A positive profit means f(x) > 0: 80x – 1,200 > 0

Add 1,200 to both sides: 80x > 1,200

Divide both sides by 80: x > 15

The smallest whole number greater than 15 is 16.

Answer: (C) 16

2.5 Graphs of Functions — Key Features

A function’s graph packs a lot of information into one picture. The SAT tests whether you can read specific features off a graph without ever seeing the function’s equation.

KEY VOCABULARY:

x-intercept (zero): a point where the graph crosses the x-axis; here f(x) = 0.
y-intercept: the point where the graph crosses the y-axis; here x = 0, so the value is f(0).
Increasing: the graph rises as x increases (moving left to right).
Decreasing: the graph falls as x increases (moving left to right).
Local maximum / minimum: a point higher (or lower) than the points immediately around it.

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SAT PATTERN ALERT To find how many solutions f(x) = k has, draw (or imagine) a horizontal line at height k and count how many times it crosses the graph. Each crossing is one solution.

Figure 3. Use this graph for Examples 2.16 through 2.20.

EXAMPLE 2.16 — Reading the y-Intercept

The graph of y = f(x) is shown above. What is the y-intercept of f?

(A) −1

(B) 0

(D) 3

The y-intercept is the point where the graph crosses the y-axis, at x = 0.

Reading the graph at x = 0 gives the point (0, −1).

Answer: (A) -1

EXAMPLE 2.17 — Counting x-Intercepts

Using the same graph of f, how many distinct x-intercepts does the function have?

(A) 2

(B) 3

(D) 5

Count how many times the graph crosses the x-axis: once between each pair of consecutive plotted points, since the y-values alternate sign the whole way across (negative, positive, negative, positive, negative).

That gives 4 sign changes, so 4 crossings.

Answer: (C) 4

EXAMPLE 2.18 — Identifying a Decreasing Interval

Using the same graph of f, on which of the following intervals is f decreasing?

(A) −4 < x < −2

(B) −2 < x < 0

(D) −1 < x < 1

A function is decreasing where the graph falls as x increases. Between x = −2 and x = 0, the graph falls from its local high point down to its local low point.

Choice (D) is a trap: it straddles the low point at x = 0, so the graph actually falls and then rises across that interval — not purely decreasing.

Answer: (B) -2 < x < 0

EXAMPLE 2.19 — Counting Solutions with a Horizontal Line

Using the same graph of f, how many values of x satisfy f(x) = 2?

(A) 0

(B) 1

(D) 3

Imagine a horizontal line at height y = 2 crossing the graph.

It crosses the first rising segment once, and the following falling segment once — 2 crossings total. The rest of the graph never reaches as high as y = 2.

Answer: (C) 2

EXAMPLE 2.20 — Finding the Maximum Value Shown

Using the same graph of f, what is the maximum value of f on the interval -4 ≤ x ≤ 4 shown?

(A) −1

(B) 1

(D) 3

Scan the graph for its highest point across the interval shown.

The highest point occurs at (−2, 3), so the maximum value of f is 3.

Answer: (D) 3

2.6 Transformations of Functions

A transformation takes the graph of a known function and shifts, reflects, or stretches it into a new graph, without changing its basic shape. The SAT tests whether you can predict the new graph — or new equation — from a description of the transformation.

TRANSFORMATION RULES:

Vertical shift: f(x) + k shifts up k units (down if k is negative)
Horizontal shift: f(x – h) shifts right h units (left if h is negative)
Reflection over the x-axis: -f(x)
Reflection over the y-axis: f(-x)
Vertical stretch/compression: a · f(x) stretches if |a| > 1, compresses if 0 < |a| < 1

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SAT PATTERN ALERT Horizontal shifts are backwards from what feels natural. f(x – 3) shifts the graph right 3 units, even though the sign inside the parentheses is negative. This is one of the most reliably tested traps in the entire Advanced Math domain.

Figure 4. Every point on f moves the same way: 3 units right and 2 units up to produce g.

EXAMPLE 2.21 — Vertical Shift — Effect on the y-Intercept

If g(x) = f(x) – 4 and the graph of f has a y-intercept of 7, what is the y-intercept of g?

(A) −4

(B) 3

(D) 11

The y-intercept of g is g(0): g(0) = f(0) – 4

Since f(0) = 7: g(0) = 7 – 4 = 3

Answer: (B) 3

EXAMPLE 2.22 — Writing a Horizontal Shift

The graph of g is the same as the graph of f, but shifted 5 units to the right. Which equation defines g(x) in terms of f(x)?

(A) g(x) = f(x − 5)

(B) g(x) = f(x + 5)

(D) g(x) = f(x) + 5

Shifting right means subtracting inside the parentheses (the input), not outside.

A shift right 5 is written as f(x − 5).

Answer: (A) g(x) = f(x – 5)

EXAMPLE 2.23 — Evaluating a Combined Transformation

If f(x) = x² and g(x) = f(x + 2) – 3, what is the value of g(1)?

(A) 3

(B) 6

(D) 12

Substitute 1 for x in g: g(1) = f(1 + 2) – 3 = f(3) – 3

Evaluate f(3) using f(x) = x²: f(3) = 9

Finish: g(1) = 9 – 3 = 6

Answer: (B) 6

EXAMPLE 2.24 — Identifying a Reflection

The graph of h(x) = -f(x) is the reflection of the graph of f over which of the following?

(A) the x-axis

(B) the y-axis

(D) the origin

Multiplying the entire output by −1 flips every y-value to its opposite, which reflects the graph over the x-axis.

Answer: (A) the x-axis

EXAMPLE 2.25 — Building an Equation from a Description

The graph of function g is obtained by shifting the graph of f(x) = |x| left 2 units and down 1 unit. What is g(x)?

(A) g(x) = |x + 2| − 1

(B) g(x) = |x − 2| − 1

(D) g(x) = |x − 2| + 1

Left 2 means replacing x with (x + 2) inside the function.

Down 1 means subtracting 1 from the entire output.

Combining both: g(x) = |x + 2| – 1

Answer: (A) g(x) = |x + 2| – 1

2.7 Composite Functions

A composite function chains two functions together: the output of one becomes the input of the other. This models real situations that happen in stages — hours worked feeding into units produced, which then feeds into profit.

COMPOSITE FUNCTION NOTATION:
(f ∘ g)(x) = f(g(x))
Read this as “f of g of x.” Work from the inside out: evaluate g first, then use that result as the input to f.

Figure 5. Composition is a two-stage assembly line — the inner function’s output becomes the outer function’s input.

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SAT PATTERN ALERT Order matters. f(g(x)) and g(f(x)) are usually different functions. Always check carefully which function is applied first — it’s the one closest to x.

EXAMPLE 2.26 — Numeric Composition

If f(x) = 2x + 1 and g(x) = x² – 3, what is the value of f(g(2))?

(A) 1

(B) 3

(D) 9

Work from the inside out. First find g(2): g(2) = 2² – 3 = 4 – 3 = 1

Now find f of that result, f(1): f(1) = 2(1) + 1 = 3

Answer: (B) 3

EXAMPLE 2.27 — Composition in the Other Order

Using the same functions f(x) = 2x + 1 and g(x) = x² – 3, what is the value of g(f(2))?

(A) 5

(B) 13

(D) 25

This time f is applied first. Find f(2): f(2) = 2(2) + 1 = 5

Now find g of that result, g(5): g(5) = 5² – 3 = 25 – 3 = 22

Notice this is a completely different result from Example 2.26 — order matters.

Answer: (C) 22

EXAMPLE 2.28 — Composition from Tables

The tables below give values of f and g for several inputs. What is the value of f(g(3))?

x	1	2	3	4
g(x)	3	1	4	2
f(x)	2	4	1	3

(A) 1

(B) 2

(D) 4

Find g(3) from the table: g(3) = 4.

Now find f(4) from the table: f(4) = 3.

Answer: (C) 3

EXAMPLE 2.29 — Algebraic Composition

If f(x) = 3x – 2 and g(x) = x + 5, what is (f ∘ g)(x)?

(A) 3x + 13

(B) 3x + 3

(D) 3x + 5

(f ∘ g)(x) means f(g(x)). Substitute the entire expression for g(x) in place of x in f: f(x + 5) = 3(x + 5) – 2

Distribute and simplify: 3x + 15 – 2 = 3x + 13

Answer: (A) 3x + 13

EXAMPLE 2.30 — Composition in a Real-World Context

A workshop’s production is modeled by g(t) = 8t, where t is hours worked and g(t) is units produced. Daily profit is modeled by f(u) = 15u – 200, where u is units produced. What is the value of (f ∘ g)(5)?

(A) 200

(B) 400

(D) 800

Find the inner function first: g(5) = 8(5) = 40 units produced.

Use that as the input to f: f(40) = 15(40) – 200

Simplify: f(40) = 600 – 200 = 400

This represents the workshop’s profit, in dollars, after 5 hours of work.

Answer: (B) 400

2.8 Chapter Formula Recap

Before the mixed practice set, use this page to review every formula and rule from the chapter in one place.

CHAPTER 2 QUICK-REFERENCE:

Function notation: f(a) = the output of f when the input is a
Domain rule (denominators): exclude any x that makes a denominator 0
Domain rule (even radicals): the radicand must be ≥ 0
Vertical shift: f(x) + k (up k, down if k < 0)
Horizontal shift: f(x – h) (right h, left if h < 0)
Reflections: -f(x) over the x-axis; f(-x) over the y-axis
Composite functions: (f ∘ g)(x) = f(g(x)) — work inside out

2.9 Mixed Practice Set — Self-Competition Round

Set a timer for 12 minutes, work all ten problems without notes, then check the Answer Key at the end. Compare each attempt only against your own previous best.

1. If f(x) = 5x – 8, what is the value of f(4)?

(A) 8

(B) 10

(D) 14

2. The table below shows values of f. For which value of x is f(x) = 9?

x	0	1	2	3
f(x)	−3	1	5	9

(A) 0

(B) 1

(D) 3

3. For the function f(x) = 7x + 5, which value of x must be excluded from the domain?

(A) −7

(B) −5

(D) 7

4. What is the domain of f(x) = √2x – 6?

(A) x ≥ 3

(B) x ≤ 3

(D) x ≥ −3

5. The value of a car is modeled by V(t) = 22,000 – 1,800t, where t is years since purchase. What does 1,800 represent?

(A) the car’s value when purchased

(B) the number of years the car has been owned

(D) the car’s value after t years

6. Using the model from item 5, after how many years will the car’s value be $9,400?

(A) 5

(B) 6

(D) 8

7. The graph of y = f(x) has x-intercepts at x = −3, x = 1, and x = 4, and no others. What is the value of f(1)?

(A) −3

(B) 0

(D) 4

8. If g(x) = f(x – 3) + 2 and the point (5, 6) lies on the graph of f, which point must lie on the graph of g?

(A) (2, 4)

(B) (8, 8)

(D) (2, 8)

9. The graph of g(x) = f(-x) is the reflection of the graph of f over which of the following?

(A) the x-axis

(B) the y-axis

(D) the line y = x

10. If f(x) = x + 6 and g(x) = 2x, what is the value of (g ∘ f)(3)?

(A) 12

(B) 15

(D) 24

Answer Key — Chapter 2 Mixed Practice

Score yourself honestly, then revisit any problem you missed. Each explanation is intentionally brief — if a solution doesn’t click immediately, trace it back to the matching worked example earlier in the chapter.

1. f(4) = 12
f(4) = 5(4) − 8 = 20 − 8 = 12.

2. x = 3
Reading the table, f(x) = 9 when x = 3.

3. x = -5
The denominator x + 5 equals 0 when x = −5.

4. x ≥ 3
2x − 6 ≥ 0 → 2x ≥ 6 → x ≥ 3.

5. the amount, in dollars, the car’s value decreases each year
1,800 is multiplied by t and subtracted, so it’s the yearly decrease in value.

6. 7
22,000 − 1,800t = 9,400 → 1,800t = 12,600 → t = 7.

7. f(1) = 0
An x-intercept means f(x) = 0 at that x-value, so f(1) = 0 directly.

8. (8, 8)
Shift right 3: 5 + 3 = 8. Shift up 2: 6 + 2 = 8. New point: (8, 8).

9. the y-axis
Replacing x with −x flips every point horizontally across the y-axis.

10. 18
f(3) = 3 + 6 = 9, then g(9) = 2(9) = 18.

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Categories

SAT Functions

Functions & Function Notation

What You Will Master in This Chapter

Table of Contents

2.1 Foundations — What Is a Function?

Is It Actually a Function? The Vertical Line Test

THE SAME IDEA IN A TABLE

2.2 Evaluating Functions

2.3 Domain and Range

RANGE, BRIEFLY

2.4 Functions in Context — Word Problems

2.5 Graphs of Functions — Key Features

2.6 Transformations of Functions

2.7 Composite Functions

2.8 Chapter Formula Recap

2.9 Mixed Practice Set — Self-Competition Round

Answer Key — Chapter 2 Mixed Practice

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Categories

What You Will Master in This Chapter

Table of Contents

2.1 Foundations — What Is a Function?

Is It Actually a Function? The Vertical Line Test

THE SAME IDEA IN A TABLE

2.2 Evaluating Functions

2.3 Domain and Range

RANGE, BRIEFLY

2.4 Functions in Context — Word Problems

2.5 Graphs of Functions — Key Features

2.6 Transformations of Functions

2.7 Composite Functions

2.8 Chapter Formula Recap

2.9 Mixed Practice Set — Self-Competition Round

Answer Key — Chapter 2 Mixed Practice

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