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Quizzes > High School Quizzes > Mathematics

Functions Worksheet Practice Quiz

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Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Colorful paper art promoting Function Frenzy trivia for high school students.

Which of the following best defines a function?
A relation where each input has exactly one output
A relation where each output has exactly one input
A mathematical procedure to perform arithmetic operations
A rule that assigns many outputs for a single input
A function is a relation that assigns exactly one output to each input. This property distinguishes functions from other types of relations.
If f(x) = 2x + 3, what is the value of f(4)?
11
8
14
10
Substitute x = 4 into the function: f(4) = 2(4) + 3 = 8 + 3 = 11. This shows the correct evaluation of the function.
What is the domain of the function f(x) = √(x - 2)?
x ≥ 2
x > 2
x ≤ 2
All real numbers
Since the square root requires a non-negative argument, the expression inside √(x - 2) must be at least 0. Thus, x - 2 ≥ 0 leads to x ≥ 2.
For the function f defined by f(x) = 3x - 5, what is f(2)?
1
6
-1
5
Evaluating the function at x = 2 gives f(2) = 3(2) - 5 = 6 - 5 = 1. This is the correct substitution and computation.
Which description best characterizes a linear function?
A straight line with constant slope
A parabola with variable curvature
A step function with sudden jumps
A sinusoidal curve that repeats periodically
A linear function has a constant rate of change, which is represented graphically by a straight line with a constant slope. This description clearly distinguishes linear functions from other types of graphs.
Given f(x) = x + 2 and g(x) = 2x, what is the value of (g ∘ f)(3)?
10
8
7
12
First, calculate f(3) = 3 + 2 = 5, then apply g to get g(5) = 2(5) = 10. This demonstrates the proper evaluation of the composite function.
If f(x) = 3x + 1, what is the inverse function f❻¹(x)?
(x - 1) / 3
3x - 1
(x + 1) / 3
3x + 1
To find the inverse, swap x and y in y = 3x + 1 and solve for y: x = 3y + 1 leads to y = (x - 1)/3. This reverses the original function's operation.
Which of the following accurately describes the vertical line test?
A curve where any vertical line intersects at most one point is a function
A graph that never intersects any vertical line
A graph where every horizontal line intersects at most one point
A graph that is symmetric about the y-axis
The vertical line test determines if a graph represents a function by checking that no vertical line intersects the graph more than once. This ensures that each input corresponds to only one output.
What is the domain of f(x) = 1/(x² - 4)?
All real numbers except 2 and -2
All real numbers
x > 2 or x < -2
x ≠ 4
The denominator x² - 4 factors to (x - 2)(x + 2) and equals zero when x = 2 or x = -2. These values must be excluded for the function to be defined.
Let f(x) = x² and g(x) = √x. What is f(g(16))?
16
8
4
2
Compute g(16) = √16 = 4, then apply f to get f(4) = 4² = 16. This sequential evaluation confirms the correct answer.
Given the piecewise function f(x) = {2x if x < 3, x² if x ≥ 3}, what is f(3)?
9
6
5
8
Since 3 meets the condition x ≥ 3, use the rule f(x) = x². Hence, f(3) = 3² = 9.
A transformation is applied to g(x) = x² to obtain f(x) = (x - 2)² + 3. Which description best explains this transformation?
Shifted 2 units to the right and 3 units upward
Shifted 2 units to the left and 3 units downward
Shifted 2 units to the right and 3 units downward
Shifted 2 units to the left and 3 units upward
The term (x - 2) indicates a horizontal shift to the right by 2 units, and the +3 indicates a vertical shift upward by 3 units. This is the correct interpretation of the transformation.
Is the function f(x) = x² one-to-one?
No, it is not one-to-one
Yes, it is one-to-one
Only for x ≥ 0
It depends on the range
A function is one-to-one if each output is produced by exactly one input. Since f(2) and f(-2) both equal 4, f(x) = x² is not one-to-one over the set of all real numbers.
How does the graph of f(x) = -3x + 5 behave?
It is a decreasing linear function with a y-intercept of 5
It is an increasing linear function with a y-intercept of 5
It is a quadratic function opening downward
It is a constant function
The slope of -3 indicates a negative, constant rate of change, meaning the function is decreasing. The constant term 5 is the y-intercept, confirming the behavior of the line.
What is the range of the linear function f(x) = 2x + 3?
All real numbers
x ≥ 3
y ≥ 3
y > 3
A linear function with a nonzero slope, like f(x) = 2x + 3, can produce any real number as an output. Therefore, its range is all real numbers.
Solve for x if f(x) = 2x - 7 and f(x) = 3.
5
3
-5
7
Setting 2x - 7 equal to 3 gives 2x - 7 = 3. Adding 7 to both sides yields 2x = 10, so x = 5. This is the correct solution.
Given f(x) = x³ - 4x and g(x) = x + 2, what is (f ∘ g)(1)?
15
11
7
9
First, calculate g(1) = 1 + 2 = 3, then substitute into f to obtain f(3) = 3³ - 4(3) = 27 - 12 = 15. This sequential process confirms the answer.
Is the function f(x) = (x - 1)² invertible over its entire domain?
No, because it is not one-to-one
Yes, it has an inverse for all real numbers
It is only invertible on a restricted domain
It is invertible when reflected across the line y = x
The function (x - 1)² produces the same output for two distinct input values, failing the one-to-one criterion. Therefore, it does not have an inverse over its entire domain.
If f(x) = 2x + 1 and g(x) = x² (with x ≥ 0), find the value of x such that f(g(x)) = 17.
2√2
4
√17
8
First, compute g(x) = x² and then f(g(x)) = 2x² + 1. Setting 2x² + 1 equal to 17 gives 2x² = 16, so x² = 8. Since x ≥ 0, x = √8 = 2√2.
Find the domain of the composite function (g ∘ f)(x), where f(x) = √(x + 4) and g(x) = 1/(x - 2).
x ≥ -4 with x ≠ 0
x > -4
x ≥ -4
All real numbers
The inner function f(x) = √(x + 4) requires x ≥ -4. The outer function g(x) becomes undefined when its input equals 2; since f(x) = 2 when x = 0, x = 0 must be excluded. Therefore, the domain is x ≥ -4 with x ≠ 0.
0
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Study Outcomes

  1. Understand function notation and evaluate functions for given inputs.
  2. Analyze function graphs to identify intercepts, slopes, and other key features.
  3. Apply transformations such as translations and dilations to function graphs.
  4. Synthesize function information to solve real-world problems.

Functions Worksheet Cheat Sheet

  1. Clear Definition of a Function - Think of a function as a magical box that assigns each input exactly one output, usually written as f(x). This rule keeps everything tidy: every x has a single partner in the output world. For instance, f(x) = x2 will always map 2 to 4, 3 to 9, and so on. Core Standards
  2. Mastering Function Notation - Function notation is the secret code for tracking inputs and outputs, with f(x) showing the output when x goes in. To evaluate, just replace x: for f(x) = 2x + 1, f(3) = 2·3 + 1 = 7. Once you get this, you'll breeze through problems like a notation ninja. Core Standards
  3. Sequences as Discrete Functions - A sequence is just a function whose domain is a list of integers, like days on a calendar. The famous Fibonacci sequence sets f(0)=1, f(1)=1, then f(n+1)=f(n)+f(n−1), rewarding you with 1, 1, 2, 3, 5, 8… endlessly. This recursive rule helps you spot patterns and predict terms down the line. Core Standards
  4. Graph Interpretation of Functions - A function's graph is like its personality portrait: intercepts show where it meets the axes, and intervals of increase or decrease reveal its mood swings. Don't forget to check end behavior to see where the graph heads as x grows big or small. Piecing these features together gives you a complete picture of relationships between quantities. Core Standards
  5. Domain in Context - The domain of a function is the set of all allowed inputs, shaped by real-world constraints. If h(n) measures hours to build n engines, n must be a positive integer - you can't build half an engine. Linking the domain to context keeps your models practical and accurate. Core Standards
  6. Average Rate of Change Demystified - Average rate of change is the slope between two points on your graph, telling how fast outputs change with inputs. It's like checking your speed over a trip: Δy/Δx gives you the "speed" of change. Mastering this concept paves the way to understanding derivatives later on. Core Standards
  7. Sketching Symbolic Functions - Plotting a symbolic function, such as f(x)=x2, reveals its shape - like a parabola with a minimum at (0,0). Identify intercepts, maxima, and minima by plugging in values or completing the square. Visualizing functions cements your intuition for how equations behave. Core Standards
  8. Equivalent Forms Reveal Secrets - Rewriting functions can highlight hidden features: factoring f(x)=x2−4 into (x−2)(x+2) shows zeros at x=±2 instantly. Completing the square or expanding also exposes vertex positions and symmetry. Switching forms is like wearing different glasses to see new details. Core Standards
  9. Comparing Functions Like a Pro - When two functions look different, compare their graphs and equations to spot which grows faster or peaks higher. For quadratics, check vertex coordinates or y‑intercepts to see which has a larger maximum. This skill helps you rank functions by behavior in any scenario. Core Standards
  10. Linear vs Exponential Growth - Linear functions, like f(x)=2x+3, add the same amount each step, while exponential functions, such as g(x)=3x, multiply by a constant factor. Recognizing which growth type you're dealing with is crucial in fields from finance to biology. Spotting the difference means you can model and predict real‑world trends accurately. Core Standards
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