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Quizzes > Mathematics

Domain and Range Quiz: Practice from Graphs, Tables, and Rules

A quick, free domain and range practice quiz with instant results.

Editorial: Review CompletedCreated By: Monica WhippleUpdated Aug 26, 2025
Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Paper art illustrating a trivia quiz on Domain and Range Mastery for high school math students.

This domain and range quiz helps you practice finding the domain and range of functions from graphs, tables, and simple rules. Get instant feedback on 20 mixed questions, with interval or set notation. For extra practice, try graphing functions practice, review domain of a function, and build skills with function notation practice.

What is the domain of the function f(x) = x + 3?
x ≥ 3
All real numbers
x > 3
x ≤ 3
Since this is a simple linear function, there are no restrictions on the value of x, so the domain is all real numbers. The other choices impose unnecessary restrictions.
What is the range of the function f(x) = 2x?
y > 0
y ≥ 0
y = 2x where x is non-negative
All real numbers
For the linear function f(x) = 2x, both the domain and range are all real numbers because there are no restrictions on the inputs or outputs. The other options incorrectly impose constraints on the range.
Given the function f(x) = √(x), what is its domain?
x > 0
x ≥ 0
x ≤ 0
All real numbers
The square root function is defined only for non-negative values of x, so the domain is x ≥ 0. The other options either exclude 0 or include negative numbers, which are not allowed.
Which notation correctly represents the set of all real numbers?
{x ∈ ℝ | x > 0}
['∞, ∞]
[0, ∞)
(-∞, ∞)
The interval (-∞, ∞) correctly represents all real numbers without restrictions, whereas the other options impose limitations or misuse notation.
What is the range of f(x) = |x|?
y ≤ 0
y ≥ 0
All real numbers
y > 0
The absolute value function produces only non-negative outputs, so the range is y ≥ 0. The other choices include negative values or incorrectly exclude zero.
Determine the domain of the function f(x) = 1/(x - 4).
(-∞, 4]
[4, ∞)
All real numbers
(-∞, 4) ∪ (4, ∞)
The function becomes undefined when the denominator is zero, which happens at x = 4. Thus, the domain excludes 4, represented by (-∞, 4) ∪ (4, ∞).
Find the domain of the function f(x) = √(2x - 6).
x > 3
All real numbers
x ≥ 3
x ≤ 3
For the square root to be defined, the radicand 2x - 6 must be non-negative. Solving 2x - 6 ≥ 0 gives x ≥ 3, which is the correct domain.
If f(x) = 1/√(x - 2), what is the domain of f?
All real numbers
x ≥ 2
x < 2
x > 2
The expression under the square root must be positive and the denominator cannot be zero, which results in the condition x > 2. The other options either include an undefined value or misrepresent the inequality.
Determine the range of the function f(x) = x².
y ≥ 0
y ≤ 0
y > 0
All real numbers
Since squaring any real number results in a non-negative value, the range of x² is y ≥ 0. The other options incorrectly include negative numbers or exclude 0.
Find the range of the function f(x) = √(9 - x²).
All real numbers
0 ≤ y ≤ 3
y ≥ 3
y > 0
The function represents the upper half of a circle with radius 3, meaning that the output values range from 0 to 3. The other options do not correctly capture this interval.
What is the domain of f(x) = √(x + 5) - √(x - 3)?
x ≥ -5
x > 3
x ≥ 3
All real numbers
Both square root expressions must be defined. While √(x + 5) allows x ≥ -5, √(x - 3) requires x ≥ 3. Thus, the domain is determined by the stricter condition: x ≥ 3.
Express the domain of f(x) = (x + 2)/(x² - 9) in interval notation.
(-∞, 3) ∪ (3, ∞)
[-3, 3]
(-∞, -3) ∪ (-3, 3) ∪ (3, ∞)
All real numbers
The denominator factors as (x - 3)(x + 3) and equals zero when x = -3 or x = 3. Excluding these values, the domain is written as (-∞, -3) ∪ (-3, 3) ∪ (3, ∞).
For the function f(x) = log(x - 1), determine the domain in interval notation.
[1, ∞)
All real numbers
(1, ∞)
(-∞, 1)
The logarithm requires its argument to be positive. Setting x - 1 > 0 results in x > 1, which in interval notation is expressed as (1, ∞).
What is the range of the function f(x) = |x - 5| + 3?
(-∞, 3]
All real numbers
(3, ∞)
[3, ∞)
Since |x - 5| is always non-negative, adding 3 shifts the minimum value to 3, making the range [3, ∞). The other choices either do not include 3 or include values that the function cannot attain.
Determine the domain of f(x) = 1/(√(x) - 2).
[0, ∞)
[4, ∞)
(0, ∞)
[0, 4) ∪ (4, ∞)
The square root function √(x) is defined for x ≥ 0; however, the denominator √(x) - 2 cannot be zero, which happens at x = 4. Therefore, the domain is [0, 4) ∪ (4, ∞).
Determine the domain and range of the function f(x) = √(4 - x)/(x - 2).
Domain: ℝ; Range: ℝ
Domain: (-∞, 4] excluding x = 2; Range: [0, ∞)
Domain: (-∞, 2) ∪ (2, 4]; Range: all real numbers
Domain: [2, 4]; Range: ℝ
The square root √(4 - x) requires x ≤ 4, and the denominator x - 2 cannot be zero, so x ≠ 2. Analyzing the behavior on the intervals (-∞, 2) and (2, 4] shows that the outputs cover all real numbers, giving the stated domain and range.
For the function f(x) = (x² - 4)/(x - 2), determine the simplified form and its domain.
f(x) = (x + 2)/(x - 2); Domain: x ≠ 2
f(x) = x - 2; Domain: all real numbers
f(x) = x² - 4; Domain: all real numbers
f(x) = x + 2; Domain: x ≠ 2
By factoring the numerator as (x - 2)(x + 2) and canceling the common factor with the denominator (while noting x ≠ 2), the function simplifies to x + 2 with the domain excluding x = 2. The alternatives do not handle the cancellation or domain correctly.
Find the domain of the piecewise function: f(x) = { x² for x < 3; √(x - 2) for x ≥ 3 }.
x ≥ 2
x < 3
x ≥ 3
All real numbers
For x < 3, the function x² is defined for every real number, and for x ≥ 3, √(x - 2) is defined because x - 2 ≥ 1. Thus, together the pieces cover all real numbers.
Determine the domain and range of f(x) = 1/(x² - 1) + √(x + 2).
Domain: {x ∈ ℝ | x ≥ -2, x ≠ -1, 1}; Range: all real numbers
Domain: all real numbers; Range: all real numbers
Domain: [-2, ∞) and Range: [0, ∞)
Domain: {x ∈ ℝ | x > -2} and Range: (0, ∞)
The square root √(x + 2) restricts x to values ≥ -2, while the denominator x² - 1 is zero when x = -1 or x = 1, which must be excluded. A deeper analysis shows the combined function can yield all real numbers as outputs.
For the function f(x) = log₂(x² - 5x + 6), determine the domain.
(-∞, 2) ∪ (3, ∞)
[2, 3]
(-∞, 2] ∪ [3, ∞)
All real numbers
The quadratic x² - 5x + 6 factors as (x - 2)(x - 3) and must be positive inside a logarithm. This inequality holds for x < 2 or x > 3, which is expressed as (-∞, 2) ∪ (3, ∞).
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Study Outcomes

  1. Identify the domain and range of given functions by analyzing their graphs and equations.
  2. Apply algebraic techniques to determine valid input and output values for functions.
  3. Evaluate function behavior at boundary points to correctly assess domain restrictions.
  4. Analyze real-world scenarios using function models to interpret domain and range implications.

Domain and Range Cheat Sheet

  1. Understand the domain of a function - Think of the domain as the "welcome mat" to your function's party: it's the set of all x-values your function will accept without crashing. For example, f(x) = 1/x politely refuses x = 0 because dividing by zero is a big no-no! Always check for any forbidden zones before you start plugging in numbers.
  2. Recognize the range of a function - The range is the collection of all y-values your function can produce - like the menu of dishes your function can serve up. For instance, f(x) = x² only cooks up non-negative results, so y is always ≥ 0. Visualizing the output helps you spot any culinary (uh, mathematical) restrictions.
  3. Identify domain and range of linear functions - Linear functions, f(x) = mx + b, are the rock stars of algebra: they march on forever in both directions. That means both domain and range span all real numbers. No matter how steep or flat the line, you've got full x- and y-coverage!
  4. Understand domain and range of quadratic functions - Quadratics f(x) = ax² + bx + c form parabolas that either open up or down, so domain is still all real numbers. But the range gets fancy: if it opens upward, y ≥ k (the vertex's y-value); if downward, y ≤ k. Finding that vertex is your golden ticket!
  5. Be aware of restrictions in rational functions - Rational functions f(x) = P(x)/Q(x) ban any x-value that makes Q(x) = 0 to avoid infinite catastrophes. Watch out for vertical asymptotes and holes, then jot down those forbidden x's. It's like marking off the landslide zones on your math map!
  6. Know the domain and range of square root functions - For f(x) = √x, the radicand (the part under the root) must be ≥ 0, so x ≥ 0. That also means your outputs are y ≥ 0 - no negative square roots in the real world! Picture the function starting at the origin and climbing gently rightward.
  7. Understand domain and range of exponential functions - Exponentials f(x) = a^x (a > 0) are the growth gurus: they take any real x, so domain is all reals, but outputs are always positive (y > 0). They never touch zero, so think of them as orbiting above the x-axis.
  8. Learn domain and range of logarithmic functions - As the inverse of exponentials, f(x) = log(x) only allows x > 0 (you can't take the log of zero or negatives). In return, it dishes out every real y-value. It's like having a backstage pass to the exponential concert!
  9. Practice with graphs - Graphs are your best friends for spotting domain and range: slide your eyes from left to right for domain, and bottom to top for range. Sketch or imagine the curve, then list all x's and y's it covers. Graphical checks can save you from algebraic slip‑ups!
  10. Master interval notation - Interval notation is the shorthand for domains and ranges, using parentheses () for exclusions and brackets [] for inclusions. For example, the domain of f(x) = 1/x is written as (-∞, 0) ∪ (0, ∞). Once you nail this, communicating sets becomes a breeze!
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