SAT — mathMedium6 min read

Domain and Range of Linear Functions

Understand domain and range for the Digital SAT. Identify valid inputs and outputs, distinguish functions from non-functions using the vertical line test.

Tutor AI Team
Tutor AI Team

Domain and range are fundamental concepts for understanding functions on the Digital SAT. The domain is the set of all possible input values (xx-values), and the range is the set of all possible output values (yy-values). The SAT tests these concepts through graphs, equations, and word problems.

For linear functions, the domain and range are typically all real numbers — but in context problems, practical constraints limit them. Understanding these concepts prepares you for both linear and nonlinear function questions.

Core Concepts

Domain

The domain of a function is the set of all values that xx can take (the valid inputs).

For the linear function f(x)=2x+3f(x) = 2x + 3:

  • Domain: all real numbers (−∞<x<∞-\infty < x < \infty)

But if xx represents "number of hours worked", the domain is x≥0x \geq 0.

Range

The range of a function is the set of all possible output values (f(x)f(x) or yy-values).

For f(x)=2x+3f(x) = 2x + 3 with domain all real numbers:

  • Range: all real numbers

But if the domain is 0≤x≤80 \leq x \leq 8, then f(0)=3f(0) = 3 and f(8)=19f(8) = 19, so range is 3≤y≤193 \leq y \leq 19.

Domain and Range from Graphs

To find the domain from a graph: look at the horizontal extent (left to right). To find the range from a graph: look at the vertical extent (bottom to top).

For a line segment from (1,3)(1, 3) to (5,11)(5, 11):

  • Domain: 1≤x≤51 \leq x \leq 5
  • Range: 3≤y≤113 \leq y \leq 11

The Vertical Line Test

A graph represents a function if and only if every vertical line crosses the graph at most once. This means each input has exactly one output.

  • A line: passes ✓ (function)
  • A circle: fails ✗ (not a function)
  • A parabola opening up/down: passes ✓ (function)
  • A parabola opening left/right: fails ✗ (not a function)

Restricted Domains

Some functions have naturally restricted domains:

  • f(x)=1x−3f(x) = \frac{1}{x-3}: domain is all real numbers except x=3x = 3 (denominator can't be 0)
  • g(x)=xg(x) = \sqrt{x}: domain is x≥0x \geq 0 (can't take square root of negative)

Linear functions with no fractions or radicals have domain = all real numbers.

Domain in Context

Word problems often restrict the domain to values that make sense:

  • Time: t≥0t \geq 0
  • Number of items: n≥0n \geq 0 and nn is an integer
  • Percentage: 0≤p≤1000 \leq p \leq 100

Strategy Tips

Tip 1: For Pure Linear Functions, Domain and Range Are All Real Numbers

Unless there's a context or restriction, f(x)=mx+bf(x) = mx + b has domain and range both equal to all real numbers (assuming m≠0m \neq 0).

Tip 2: For Graphs, Read the Endpoints

Look for where the graph starts and stops. Closed circles (●) mean the endpoint is included; open circles (○) mean it's excluded.

Tip 3: For Context Problems, Think About What Makes Sense

You can't have negative time, negative people, or fractional items (usually). These constraints define the domain.

Tip 4: Range Depends on Domain

Once you know the domain, evaluate the function at the domain boundaries to find the range.

Tip 5: On the SAT, Domain Questions Often Involve Denominators or Square Roots

Look for values that make a denominator zero or a radicand negative.

Worked Example: Example 1

Problem

What is the domain of f(x)=5x+2f(x) = \frac{5}{x + 2}?

x+2≠0x + 2 \neq 0 → x≠−2x \neq -2

Domain: all real numbers except x=−2x = -2.

Solution

Worked Example: Example 2

Problem

What is the domain of g(x)=3x−6g(x) = \sqrt{3x - 6}?

3x−6≥03x - 6 \geq 0 → x≥2x \geq 2

Domain: x≥2x \geq 2.

Solution

Worked Example: Example 3

Problem

A graph shows a line segment from (−2,5)(-2, 5) to (4,−1)(4, -1). What is the range?

The y-values go from −1-1 to 55.

Range: −1≤y≤5-1 \leq y \leq 5.

Solution

Worked Example: SAT-Style Context

Problem

A parking garage charges C(h)=5h+3C(h) = 5h + 3 dollars for hh hours of parking, where 0≤h≤240 \leq h \leq 24. What is the range of CC?

C(0)=3C(0) = 3 and C(24)=123C(24) = 123

Range: 3≤C≤1233 \leq C \leq 123.

Solution

Worked Example: Example 5

Problem

Does the equation x2+y2=25x^2 + y^2 = 25 represent yy as a function of xx?

No — for most xx-values, there are two yy-values (e.g., when x=0x = 0, y=5y = 5 or y=−5y = -5). It fails the vertical line test.

Solution

Practice Problems

  1. Problem 1

    What is the domain of f(x)=xx−5f(x) = \frac{x}{x - 5}?

    Problem 2

    What is the domain and range of the function shown as a line segment from (0,2)(0, 2) to (6,8)(6, 8)?

    Problem 3

    A company models profit as P(n)=12n−500P(n) = 12n - 500 for nn items sold (n≥0n \geq 0). What is the domain? What is the minimum of the range?

    Problem 4

    What is the domain of h(x)=10−2xh(x) = \sqrt{10 - 2x}?

    Problem 5

    Does y=∣x∣y = |x| pass the vertical line test? What is its range?

    Problem 6

    If f(x)=−3x+15f(x) = -3x + 15 and the domain is 1≤x≤51 \leq x \leq 5, what is the range?

Want to check your answers and get step-by-step solutions?

Get it on Google PlayDownload on the App Store

Common Mistakes

  • Confusing domain and range. Domain = x-values (inputs); range = y-values (outputs).
  • Forgetting context constraints. In real-world problems, negative values may not be valid even if the math allows them.
  • Missing excluded values. For 1x−3\frac{1}{x-3}, the domain excludes x=3x = 3, not x=−3x = -3.
  • Reversing inequality for range. If the function is decreasing, a larger xx gives a smaller yy. Be careful when stating the range.
  • Not checking open vs. closed endpoints on graphs. Open circle = excluded; closed circle = included.

Frequently Asked Questions

Do all linear functions have the same domain and range?

All non-constant linear functions (m≠0m \neq 0) have domain and range both equal to all real numbers. A constant function f(x)=cf(x) = c has domain all real numbers but range {c}\{c\}.

Can a function's range be a single number?

Yes — for example, f(x)=5f(x) = 5 (a constant function) has range {5}\{5\}.

How do I express domain and range?

Use inequality notation (x≥2x \geq 2), interval notation ([2,∞)[2, \infty)), or set notation ({x:x≥2}\{x : x \geq 2\}). The SAT typically uses inequality notation.

Is the domain always given in context problems?

Sometimes explicitly, sometimes you need to infer it from the situation.

Can a linear function have a restricted range without a restricted domain?

Only for constant functions (m=0m = 0). For m≠0m \neq 0, restricting the range automatically restricts the domain.

Key Takeaways

  • ✓

    Domain = valid inputs (xx-values); range = resulting outputs (yy-values).

  • ✓

    Unrestricted linear functions have domain and range of all real numbers.

  • ✓

    Context restricts domain (e.g., time ≥ 0, whole numbers only).

  • ✓

    From graphs: domain = horizontal extent, range = vertical extent.

  • ✓

    Vertical line test determines if a graph represents a function.

  • ✓

    Denominators ≠ 0 and radicands ≥ 0 create domain restrictions.

Tags:
#sat#math#algebra#functions#domain#range

Related Topics

Ready to Ace Your SAT math?

Get instant step-by-step solutions to any problem. Snap a photo and learn with Tutor AI — your personal exam prep companion.

Get it on Google PlayDownload on the App Store