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Quizzes > High School Quizzes > Mathematics

Algebra 1 Unit 3 Practice Quiz

Master Relations and Functions with Key Answers

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art promoting Algebra 1 Function Focus practice quiz for high school students.

Which of the following best defines a function?
A rule that assigns each input exactly two outputs.
A relation where each input has exactly one output.
A set of numbers that increases as the input increases.
A process that sometimes assigns more than one output to an input.
A function is defined as a relation in which every input is associated with exactly one output. This property distinguishes functions from general relations.
What is the domain of the function f(x) = 2x + 3?
All real numbers.
All positive numbers.
All integers.
Only numbers greater than 3.
The function f(x) = 2x + 3 is defined for every real number because there are no restrictions on x. Therefore, the domain is all real numbers.
What is the range of the constant function f(x) = 5?
{5}
All real numbers.
[5, ∞)
(-∞, 5)
A constant function returns the same value for every input. Hence, its range is the single value {5}.
What is the value of f(2) if f(x) = x² + 1?
3
4
5
6
Substitute x = 2 into the function: f(2) = 2² + 1 = 4 + 1 = 5. Therefore, the correct answer is 5.
If f(x) = 10 for all x, what is f(5)?
10
5
0
Undefined
A constant function outputs the same value regardless of the input. Thus, f(5) equals 10.
Which of the following sets can be the domain of the function f(x) = 1/(x - 3)?
All real numbers except 3.
All real numbers except 1/3.
All real numbers.
Only numbers greater than 3.
The function f(x) = 1/(x - 3) is undefined when the denominator is zero, which happens at x = 3. Thus, the domain is all real numbers except 3.
Given f(x) = 2x - 1 and g(x) = x², what is (g ∘ f)(2)?
9
16
25
7
First, compute f(2) = 2(2) - 1 = 3, then compute g(3) = 3² = 9. Therefore, (g ∘ f)(2) equals 9.
If h(x) = √(x - 2), what is h(11)?
3
√11
9
5
Replacing x with 11, we have h(11) = √(11 - 2) = √9, which simplifies to 3.
Which of the following functions is even?
f(x) = x³
f(x) = x²
f(x) = x + 2
f(x) = 2ˣ
An even function satisfies f(x) = f(-x) for all x. The function f(x) = x² meets this criterion, making it an even function.
If f(x) = x + 4 and g(x) = 3x, what is (f ∘ g)(x)?
3x + 4
3x + 12
x + 7
3x - 4
Calculating the composite function: f(g(x)) = (3x) + 4, which simplifies to 3x + 4. This is the correct expression for the composite function.
Which of the following graphs violates the vertical line test?
The graph of y = x².
The graph of a circle defined by x² + y² = 16.
The graph of a line with slope 2.
The graph of a parabola opening upward.
The vertical line test determines if a graph represents a function. A circle is intersected by some vertical lines at more than one point, hence it fails the test.
What is the inverse of the function f(x) = x + 7?
f❻¹(x) = x - 7
f❻¹(x) = x + 7
f❻¹(x) = 7 - x
f❻¹(x) = -x + 7
To find the inverse, swap x and y in the equation y = x + 7 and solve for y. The resulting inverse function is f❻¹(x) = x - 7.
Which method is not used to describe a function?
A description where one input corresponds to multiple outputs.
A mapping diagram.
A set of ordered pairs.
A table of values.
A function must assign exactly one output to each input. A method that maps one input to multiple outputs does not correctly describe a function.
Consider the function f(x) = |x - 2|. Which description best fits its graph?
A parabola opening upwards.
A V-shaped graph with vertex at (2, 0).
A horizontal line.
A straight line with a positive slope.
The graph of an absolute value function is V-shaped. The function |x - 2| shifts the vertex to (2, 0), distinguishing it from parabolic or linear graphs.
Let f(x) = (2x + 3)/(x - 1). What is f(2)?
7
5
Undefined
10
By substituting x = 2, we get f(2) = (2(2) + 3)/(2 - 1) = (4 + 3)/1 = 7. Thus, the correct answer is 7.
Given f(x) = √(x + 6) and g(x) = x², what is (f ∘ g)(-2)?
√10
10
4
-√10
First compute g(-2): (-2)² = 4. Then, f(4) = √(4 + 6) = √10. Hence, (f ∘ g)(-2) equals √10.
Determine the inverse of the function f(x) = (x - 5)/3.
f❻¹(x) = 3x + 5
f❻¹(x) = 3x - 5
f❻¹(x) = (x + 5)/3
f❻¹(x) = (x - 5)/3
Start with y = (x - 5)/3, then solve for x to get x = 3y + 5. Swapping variables yields the inverse function f❻¹(x) = 3x + 5.
For the piecewise function f(x) defined as f(x) = x² for x < 0 and f(x) = 2x + 1 for x ≥ 0, what are the values of f(-3) and f(2)?
f(-3) = 9, f(2) = 5
f(-3) = -9, f(2) = 5
f(-3) = 9, f(2) = 4
f(-3) = -3, f(2) = 5
For x < 0, use f(x) = x², so f(-3) = (-3)² = 9. For x ≥ 0, use f(x) = 2x + 1, giving f(2) = 2(2) + 1 = 5.
Find the composite function (g ∘ f)(x) where f(x) = x - 2 and g(x) = 1/(x + 1).
1/(x - 1)
1/(x + 1)
1/(x - 3)
(x - 2)/(x + 1)
To compute (g ∘ f)(x), substitute f(x) into g(x): g(x - 2) = 1/((x - 2) + 1) = 1/(x - 1). This is the simplified composite function.
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Study Outcomes

  1. Analyze the structure and properties of functions, including domain and range.
  2. Evaluate function notation and compute function outputs for given inputs.
  3. Interpret graphical representations of functions to identify key characteristics.
  4. Apply problem-solving strategies to function-related questions and scenarios.
  5. Synthesize relationships between variables by constructing and analyzing functional models.

Algebra 1 & 2 Unit Test Answer Keys Cheat Sheet

  1. Understand the Definition of a Function - A function is like a one-to-one pen pal system: each input gets exactly one output, and no input is left hanging. For instance, the set {(1,2), (2,3), (3,4)} shows each x paired up perfectly with one y. This clarity makes graphing a breeze and avoids messy multi-valued disasters! Function Definition Tutorial
  2. Master Function Notation - Function notation (f(x)) is your superhero cape for tackling math problems, showing clearly which function you're calling on. If f(x) = x², then f(3) = 9 instantly tells you 3 gets squared to give 9. Grasping this notation means you can confidently evaluate and manipulate functions without breaking a sweat. Evaluating Functions Practice
  3. Learn to Evaluate Functions - Plugging in numbers into function rules is like feeding the machine the right ingredients to get your expected output. For g(x) = 2x + 5, feeding in 4 yields g(4) = 13 - simple cooking! Regular practice makes these substitutions feel as natural as your daily coffee fix. Function Evaluation Practice
  4. Identify Domain and Range - Think of the domain as the guest list (all x-values allowed) and the range as the party pictures (all y-values showing up). For f(x) = √x, only x ≥ 0 get an invite, and the photos (y) are also non-negative. Spotting these restrictions helps you avoid imaginary or undefined scenarios. SparkNotes: Domain & Range
  5. Recognize Different Function Types - Whether you're cruising on linear roads (y = mx + b), looping through parabolic hills (y = ax² + bx + c), or skyrocketing with exponential blast-offs (y = a·b^x), each function has its unique vibe. Linear graphs keep it consistent, quadratics add drama with curves, and exponentials grow like wildfire. Knowing their signatures helps you predict behavior at a glance! Common Functions Guide
  6. Understand Function Transformations - Shifting, stretching, and flipping functions is like remixing your favorite tune: f(x) + k bumps it up k units, f(x - h) slides it right by h, and - f(x) gives it a vertical mirror spin. Master these moves and you'll graph in record time without missing a beat. Function Transformations
  7. Practice Operations with Functions - Adding, subtracting, multiplying, and dividing functions is function fusion at its finest. If f(x) = x + 1 and g(x) = x², then (f + g)(x) = x² + x + 1, instantly combining powers and linear flair. These combos are key ingredients for tackling multi-layered problems like a pro chef. Function Operations Practice
  8. Learn About Piecewise Functions - Piecewise functions are like mood rings for math, switching formulas across intervals to reflect different behaviors. For example, f(x) might be x² when x < 0 but x + 2 when x ≥ 0, adapting to each region's rules. They're perfect for modeling real-life scenarios with changing conditions. Piecewise Functions Guide
  9. Understand Inverse Functions - Inverse functions are the ultimate undo button, reversing the effects of the original function so f(f❻¹(x)) = x. If f(x) = 2x + 3, solving for f❻¹(x) lets you swap inputs and outputs like a smooth swap meet. This skill is a lifesaver when you need to solve equations and reverse-engineer results. Inverse Functions Guide
  10. Apply Functions to Real-World Problems - Functions let you turn practical scenarios - like distance traveled (d(t) = speed × time) - into neat mathematical models. Want to predict cool-down rates, population growth, or budget forecasts? Functions are your backstage pass. Connecting these equations to everyday life makes math feel like a superpower! Real-World Functions Examples
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