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Quizzes > High School Quizzes > Mathematics

Practice Quiz: Grade 11 Relations & Functions

Master key concepts with engaging practice problems

Difficulty: Moderate
Grade: Grade 11
Study OutcomesCheat Sheet
Colorful paper art promoting Relations and Functions Rally, a math trivia quiz for high school students.

Which of the following best defines a function?
A relation in which outputs are functions of the inputs with no restrictions.
A relation that maps one output to multiple inputs simultaneously.
A relation that contains only ordered pairs with the same elements repeated.
A relation in which every input is paired with exactly one output.
A function is defined as a relation where each input is paired with exactly one output. This unique pairing is essential to differentiate functions from other types of relations.
What does the domain of a function represent?
The rule that assigns outputs to inputs.
The x-intercept of the graph representing the function.
The set of all outputs of the function.
The set of all possible inputs.
The domain of a function is the complete set of values for which the function is defined. It specifies which inputs are allowed, making option A the correct answer.
What is the range of a function?
The set of all outputs produced by the function.
The set of all inputs accepted by the function.
The values at which the function is not defined.
The list of intervals where the function increases.
The range refers to the set of all possible outputs of a function. Since it includes every output that results from the allowed inputs, option A is correct.
How can you determine if a graph represents a function?
By verifying if the graph is symmetrical about the y-axis.
By checking if the graph has horizontal lines.
By applying the vertical line test.
By ensuring that every horizontal line crosses the graph at most once.
The vertical line test is a standard method used to determine if a graph represents a function. If any vertical line intersects the graph more than once, the graph fails to represent a function.
Which of the following sets of ordered pairs represents a valid function?
{(1, 2), (3, 4), (5, 6)}
{(7, 8), (9, 10), (7, 8)}
{(2, 3), (4, 5), (2, 6)}
{(1, 5), (1, 5), (2, 6)}
A valid function must assign exactly one output to each input value. In the provided options, only the set {(1, 2), (3, 4), (5, 6)} meets this criterion.
Consider the function f(x) = 2x + 3. What is f(4)?
7
14
11
8
Substituting x = 4 into the function gives f(4) = 2(4) + 3, which equals 8 + 3 = 11. This makes option A the correct answer.
Which of the following graphs violates the definition of a function?
A parabola opening upwards
A circle
A hyperbola
A straight line
A circle does not pass the vertical line test since many vertical lines intersect it in two points. This violation disqualifies it as a function, making option A correct.
If f(x) = x², what is the domain of f when considered as a function from real numbers to real numbers?
Non-negative real numbers
Positive integers
All positive real numbers
All real numbers
The function f(x) = x² is defined for every real number, meaning its domain includes all real numbers. This makes option A the correct choice.
For the function f(x) = 1/(x - 2), which value of x must be excluded from the domain?
x = 1
x = 2
x = -2
x = 0
Since the denominator becomes zero when x = 2, the function is undefined at that point. Therefore, x = 2 must be excluded from the domain, making option A correct.
In a mapping diagram, if input a maps to output b and input c also maps to output b, what property is being demonstrated?
Neither injectivity nor surjectivity necessarily
Surjectivity
Injectivity
It is a function but not injective
When different inputs map to the same output, the function fails to be one-to-one (injective) while still satisfying the definition of a function. Thus, option D, indicating a function that is not injective, is correct.
Given the relation R = {(1, 4), (2, 5), (3, 4), (2, 6)}, why is R not a function?
Because it has negative outputs.
Because the input 4 corresponds to two different outputs.
Because the relation does not include a zero input.
Because the input 2 corresponds to both 5 and 6.
A key requirement for a relation to be a function is that each input must map to exactly one output. Since input 2 maps to both 5 and 6, the relation violates this rule, making option A the correct answer.
What is the inverse function of f(x) = 3x - 7, assuming it is invertible?
f❻¹(x) = (x - 7)/3
f❻¹(x) = (x + 7)/3
f❻¹(x) = 3/(x - 7)
f❻¹(x) = (7 - x)/3
To find the inverse of a function, swap x and y in the equation and solve for y. Carrying out this process for f(x) = 3x - 7 results in f❻¹(x) = (x + 7)/3, which is why option A is correct.
If f(x) = √(x - 1), what is the domain of f?
x ≥ 1
x ≤ 1
All real numbers
x > 1
The expression under the square root must be non-negative. Setting x - 1 ≥ 0 gives x ≥ 1, which makes option A the correct answer.
Which of the following statements is true about composite functions?
Composite functions always result in a polynomial function.
The composite f ◦ g means applying g first, then applying f.
A composite function reverses the order of inputs and outputs.
The composite of two functions is always commutative.
Composite functions involve applying one function to the result of another. Specifically, (f ◦ g)(x) means f(g(x)), so option B, which describes applying g first then f, is correct.
What is the output of the function f(x) = |x - 3| when x = 2?
3
2
1
0
Substituting x = 2 into f(x) gives |2 - 3| = | -1 |, which simplifies to 1. This makes option A the correct answer.
If f(x) = 2x/(x - 1) and g(x) = x + 3, what is (f ◦ g)(x)?
(x + 3)/(x + 2)
2(x + 3)/(x + 2)
(2x + 6)/(x + 3)
2(x + 3)/(x - 2)
To form the composite function f(g(x)), substitute g(x) = x + 3 into f(x) to get f(x + 3) = 2(x + 3)/((x + 3) - 1). Simplifying the denominator gives x + 2, so the composite function is 2(x + 3)/(x + 2).
Determine the inverse of the function f(x) = (x - 4)/(2x + 3), assuming it is invertible.
f❻¹(x) = (3x - 4)/(2x + 1)
f❻¹(x) = (4 + 3x)/(2x + 1)
f❻¹(x) = (-3x - 4)/(2x - 1)
f❻¹(x) = (3x + 4)/(1 - 2x)
To find the inverse, switch x and y in the equation y = (x - 4)/(2x + 3) and solve for y. After rearranging and solving, the inverse is found to be f❻¹(x) = (-3x - 4)/(2x - 1), making option A correct.
For the piecewise function defined as f(x) = { x² if x ≤ 2, 3x + 1 if x > 2 }, what are the values of f(2) and f(3) respectively?
4 and 7
4 and 10
2 and 7
2 and 10
For x = 2, since 2 is within the domain of the first piece, f(2) = 2² = 4. For x = 3, using the second piece gives f(3) = 3(3) + 1 = 10, so option A is correct.
Consider f(x) = 1/x (with x ≠ 0) and g(x) = x + 2. What is the domain of the composite function (f ◦ g)(x)?
All real numbers except -2
x < -2
All real numbers except 0
x > -2
The composite function (f ◦ g)(x) becomes 1/(x + 2). It is undefined when the denominator is zero, which happens when x = -2. Therefore, the domain excludes -2, making option A correct.
Let f(x) = 2x + 5 and g(x) = x². Which of the following represents (g ◦ f)(x)?
2x² + 10x + 5
(2x + 5)²
4x² + 10x + 25
2x² + 5
The composite function (g ◦ f)(x) is defined as g(f(x)). Substituting f(x) = 2x + 5 into g(x) yields (2x + 5)². This confirms that option A is the correct representation.
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Study Outcomes

  1. Analyze relation properties to distinguish between functions and non-functions.
  2. Evaluate the domain and range of given relations and functions.
  3. Apply function notation to solve mathematical problems.
  4. Determine inverse functions and examine their characteristics.
  5. Interpret real-world scenarios using functions to model relationships.

Relations & Functions Worksheet Grade 11 Cheat Sheet

  1. Relation vs. Function - A relation is simply a set of ordered pairs, but a function is a special relation where each input leads to exactly one output. Think of it like a vending machine: you press one button and you get one snack - no surprises! This fundamental distinction helps you predict and work with mappings confidently. OpenStax: Relations & Functions
  2. Function Notation - Using f(x) makes talking about functions super clean - f(3) = 9 just tells you what happens when x is 3. It's like a shortcut for "when I plug in this number, what comes out?" Getting comfortable with f(x), g(t), or h(z) sets you up for everything from graphing to calculus. Symbolab: Key Concepts
  3. Domain and Range - The domain is all possible x-values you're allowed to use, and the range is every y-value you can get out. For example, f(x)=√x only works for x ≥ 0, so that's your domain. Mastering this means you'll never accidentally plug in a forbidden number again! OpenStax: Relations & Functions
  4. Vertical Line Test - Draw a vertical line through your graph - if it ever hits the curve more than once, you've got a relation that's not a function. It's like making sure each seat at a one-person table gets only one guest. This quick visual trick saves tons of time on exams. Higher Math Made Simple
  5. One-to-One Functions - In a one-to-one function, every output comes from a unique input - no two inputs share the same result. This property is essential when you want to find an inverse, because you need to "undo" the function without confusion. Higher Math Made Simple
  6. Evaluating Functions - To evaluate, just replace x with your number and follow the steps - if f(x)=2x+3, then f(4)=2(4)+3=11. Treat each substitution like a puzzle: plug, simplify, and solve for that satisfying "aha" moment. Symbolab: Key Concepts
  7. Types of Functions - Get to know linear (straight lines), quadratic (parabolas), and absolute value (big Vs) inside and out. Each has its own graph shape and rules, so recognizing them instantly will level up your graph-sketching game. Symbolab: Key Concepts
  8. Inverse Functions - If f(g(x))=x and g(f(x))=x, then f and g reverse each other's steps. Finding an inverse often means swapping x and y and solving for the new y - like rewinding a movie to go back to the start. BYJU'S: Relations & Functions
  9. Composite Functions - Combining two functions is like stacking LEGO blocks: (f ∘ g)(x)=f(g(x)). For instance, g adds 1, then f squares that result - perfect practice for multi-step problems. Order matters here, so watch your parentheses! Careers360: Chapter on Relations & Functions
  10. Real-World Modeling - Functions pop up everywhere: predicting population growth, calculating distances, or even modeling your favorite video game physics. Seeing functions in action makes the math click - and turns abstract symbols into practical superpowers! SchoolTube: Understanding Relations & Functions
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