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Quizzes > High School Quizzes > Mathematics

Unit 2 Functions Practice Test

Ace your quiz with a detailed answer key

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art depicting a trivia quiz on function fundamentals for high school students.

Which of the following best defines a function in mathematics?
Every input is paired with exactly one output.
Each input is mapped to multiple outputs.
There is no restriction on input-output pairings.
Every output is paired with exactly one input.
A function is a relation where each input is assigned exactly one output. This distinct property separates functions from general relations.
In the notation f(x), what does 'x' typically represent?
The function's output.
The input or independent variable.
The slope of the function.
The function's name.
In function notation, 'x' is the independent variable that is input into the function. This value is subsequently used to calculate the corresponding output f(x).
Which of the following is true about the domain of a function?
The domain refers to the function's slope.
The domain consists of all output values.
The domain is always the set of all real numbers.
The domain consists of all input values for which the function is defined.
The domain of a function includes all values that can be input without causing the function to be undefined. It is a fundamental concept ensuring the proper operation of the function.
What is the horizontal line test used to determine?
Whether a function has an inverse.
Whether the range is limited to nonnegative numbers.
Whether a function is linear.
Whether the graph represents a parabola.
The horizontal line test checks if any horizontal line intersects the graph more than once. If it does, then the function is not one-to-one and does not have an inverse.
Which of the following describes a constant function?
A function where the output equals the input.
A function that increases as the input increases.
A function that decreases as the input increases.
A function where the output remains the same regardless of the input.
A constant function yields the same output for every input, which results in a horizontal line on its graph. This property clearly distinguishes it from other types of functions where outputs vary.
Given the function f(x) = 2x + 5, what is f(3)?
10
11
6
8
Substitute x with 3 into the function f(x)=2x+5 to get 2*3+5, which equals 11. This direct evaluation reinforces the process of function substitution.
How is the graph of f(x) = (x - 2)^2 related to the graph of f(x) = x^2?
It is shifted 2 units to the right.
It is vertically stretched by a factor of 2.
It is reflected over the x-axis.
It is shifted 2 units to the left.
Replacing x with (x - 2) in the function results in a horizontal shift to the right by 2 units. This transformation is a common example of horizontal translation in function graphs.
If f(x) = x + 3 and g(x) = 2x, what is the value of (g - f)(2)?
10
9
8
7
First calculate f(2) by adding 3 to 2, which gives 5. Then, apply g to this result: g(5) equals 2*5, which is 10.
Which of the following functions has an inverse?
f(x) = x²
f(x) = |x|
f(x) = 3x - 4
f(x) = x² + 5
A function must be one-to-one to possess an inverse. Linear functions like f(x)=3x-4 meet this requirement, while quadratic and absolute value functions typically do not when defined over all real numbers.
What is the y-intercept of the function f(x) = -4x + 7?
7
-4
-7
4
The y-intercept is found by setting x to 0 in the function. Here, f(0) = -4(0) + 7 equals 7, making 7 the y-intercept.
What is the range of the function f(x) = x² for all real numbers x?
All real numbers
f(x) ≥ 0
f(x) ≤ 0
f(x) > 0
Since any real number squared is nonnegative, the output of f(x)=x² is always greater than or equal to zero. This set of nonnegative numbers is the function's range.
How does the graph of f(x) = |x| change when it is modified to f(x) = |x - 3|?
It shifts 3 units to the right.
It shifts 3 units to the left.
It shifts 3 units upward.
It reflects over the y-axis.
The expression |x - 3| indicates that the input is adjusted by subtracting 3, which shifts the entire graph horizontally to the right by 3 units. This is a common horizontal translation in absolute value functions.
For functions f(x) and g(x), which statement best describes the composition (f - g)(x)?
It represents the product of f(x) and g(x).
It means f(g(x)).
It represents the sum of f(x) and g(x).
It means g(f(x)).
The notation (f - g)(x) stands for f(g(x)), meaning that g(x) is calculated first and its value is then used as the input for f. This is the standard method for composing functions.
If h(x) = 2x - 1, what is the first step in finding its inverse, h❻¹(x)?
Swap x and y and then solve for y.
Multiply the entire function by -1.
Add 1 to both sides to isolate x.
Replace x with 1/x and solve for y.
To determine the inverse of a function, the standard procedure is to exchange the roles of x and y. This swap is followed by solving the resulting equation for y, which yields h❻¹(x).
Which transformation occurs when the function f(x) = √x is rewritten as f(x) = √(x + 4)?
The graph shifts 4 units upward.
The graph shifts 4 units to the right.
The graph is reflected across the y-axis.
The graph shifts 4 units to the left.
Adding 4 inside the square root function (x + 4) causes the graph to shift horizontally to the left by 4 units. This internal modification of the input is a classic example of a horizontal translation.
Consider the piecewise function defined as f(x) = { x + 2 for x < 0, x² for x ≥ 0 }. What are the values of f(-3) and f(2), respectively?
f(-3) = -1 and f(2) = 4
f(-3) = 9 and f(2) = 5
f(-3) = -1 and f(2) = 5
f(-3) = 9 and f(2) = 4
For values of x less than 0, the function f(x) is defined as x + 2, so f(-3) equals -1. For x greater than or equal to 0, the function is given by x², yielding f(2) = 4.
Which of the following statements about function composition is always true if both f and g are invertible functions?
(f - g)❻¹ = g❻¹ - f❻¹
(f - g)❻¹ = g - f
(f - g)❻¹ = f - g
(f - g)❻¹ = f❻¹ - g❻¹
When two functions f and g are invertible, the inverse of their composition is obtained by composing their inverses in reverse order. This result is a fundamental property of invertible functions.
When graphing the function f(x) = |2x - 4|, what effect does the multiplier 2 inside the absolute value have on the graph?
It causes a vertical compression by a factor of 1/2.
It causes a horizontal stretch by a factor of 2.
It causes a vertical stretch by a factor of 2.
It causes a horizontal compression by a factor of 1/2.
The coefficient 2 multiplying x inside the absolute value affects the horizontal scale of the graph. Specifically, it compresses the graph horizontally by a factor of 1/2, altering how quickly the graph changes.
Find the inverse of the function f(x) = (x - 5)/3.
f❻¹(x) = 3x - 5
f❻¹(x) = (x + 5)/3
f❻¹(x) = (x - 5)/3
f❻¹(x) = 3x + 5
To find the inverse, start with y = (x - 5)/3, swap x and y to get x = (y - 5)/3, and then solve for y, which results in y = 3x + 5. This procedure is the standard method for finding the inverse of a function.
Which of the following describes the transformation of f(x) when it is changed to f(2x) + 3?
The graph is vertically stretched by a factor of 2 and shifted upward by 3 units.
The graph is horizontally stretched by a factor of 2 and shifted upward by 3 units.
The graph is vertically compressed by a factor of 1/2 and shifted upward by 3 units.
The graph is horizontally compressed by a factor of 1/2 and shifted upward by 3 units.
Multiplying the input variable by 2 produces a horizontal compression by a factor of 1/2, while adding 3 to the function shifts the graph upward by 3 units. This question combines two common transformation concepts into one analysis.
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Study Outcomes

  1. Identify and interpret function notation accurately.
  2. Analyze graphs of functions to determine key characteristics.
  3. Apply function transformations to modify and sketch graphs.
  4. Evaluate how changes in function expressions affect their graphical representation.
  5. Synthesize information from notation and graphs to solve problem-based function questions.

Unit 2 Understanding Functions Answer Key Cheat Sheet

  1. Function Notation - Think of f(x) as a magic machine: you feed it x and it spits out a result. For example, if f(x)=x², then f(3)=9 - simple as that! Grasping this notation unlocks all sorts of function wizardry. Transformations of Functions - MathBitsNotebook(A2)
  2. Graphing Functions - Plotting a function is like connecting the dots on a treasure map: substitute x values, find the points, and draw a smooth curve. For instance, with f(x)=x² you'd mark (0,0), (1,1), and (-1,1). A clear graph reveals hidden behavior at a glance! Transformations of Functions - MathBitsNotebook(A2)
  3. Vertical Shifts - Adding or subtracting a constant k to f(x) moves your curve up or down like sliding it on a ruler. So f(x)+3 nudges everything 3 units skyward, while f(x)−2 drops it down. Mastering this helps you predict how graphs change. Transformations of Functions - MathBitsNotebook(A2)
  4. Horizontal Shifts - Replace x with (x−h) to move the graph h units right, or with (x+h) to shift it left. For example, f(x−2) scoots the entire curve 2 steps right on the x-axis. It's like panning the camera left or right! Transformations of Functions - MathBitsNotebook(A2)
  5. Reflections - Multiply f(x) by −1 to flip the graph over the x-axis, turning peaks into valleys. Swap x for −x to mirror over the y-axis, flipping it sideways. It's a fun way to see how functions behave when mirrored! Transformations of Functions - MathBitsNotebook(A2)
  6. Vertical Stretches & Compressions - Multiply f(x) by a factor a>1 to stretch the graph tall or by 0Transformations of Functions - MathBitsNotebook(A2)
  7. Horizontal Stretches & Compressions - Change x to (x/b) to stretch the graph wide if b>1 or compress it if 0Transformations of Functions - MathBitsNotebook(A2)
  8. Combining Transformations - Stack transformations like a sandwich: f(x−2)+3 shifts right 2 units then up 3 units in one go. The order matters - like mixing colors, you'll get different results. Practice layering moves to ace complex graphs! Practice: Combining Function Transformations - MathBitsNotebook(A1)
  9. Identifying Transformations - Given g(x)=−2(x+1)²+4, pinpoint the flip, stretch, and shifts: reflection, vertical stretch, left shift, and upward move. Spotting each change makes graphing quick and intuitive. Test yourself with real examples! Practice Identifying Function Transformations - MathBitsNotebook(A1)
  10. Online Practice Resources - Dive into interactive lessons, video tutorials, and quizzes to reinforce everything you've learned about function transformations. Consistent practice builds confidence and speed. Level up your skills with top-quality resources! Transformations Explained: Definition, Examples, Practice & Video Lessons
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