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Quizzes > High School Quizzes > Mathematics

Practice Quiz: Characteristics of Functions Part 1

Boost your skills with targeted review questions

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Paper art promoting Functions in Focus quiz for high school students to improve math skills.

Which of the following best defines a function?
A rule that assigns each input exactly one output
A type of equation with one variable
A process that only uses positive numbers
A relation that assigns one output to many inputs
A function uniquely pairs each input with a single output, which is essential to its definition. This distinguishes functions from general relations that may assign multiple outputs to one input.
Which test is used to determine if a graph represents a function?
Diagonal line test
Vertical line test
Horizontal line test
Line symmetry test
The vertical line test confirms that a graph represents a function by ensuring no vertical line intersects the graph more than once. This is the standard method used to verify the definition of a function.
Identify which relation can be considered a function.
y = 2x + 3
x^2 + y^2 = 1
x = y^2
y^2 = x
The equation y = 2x + 3 represents a linear function where each input corresponds to one unique output. The other relations fail the vertical line test or involve ambiguous mappings.
What is the domain of a function?
The x-intercepts of the graph
All possible output values
All possible input values
The set of solutions to the equation
The domain of a function is the complete set of input values for which the function is defined. It is fundamental in understanding where the function operates.
Which of these is a characteristic of a well-defined function?
Inputs must be positive numbers
Each output corresponds to only one input
Each input has exactly one output
The relation must have at least one output
A well-defined function guarantees that every input is paired with precisely one output, ensuring no ambiguity. This is the primary criterion that distinguishes functions from other types of relations.
Given the function f(x) = x^2, what is the range of f?
y ≤ 0
All real numbers
y ≥ 0
x ≥ 0
Squaring any real number yields a nonnegative result, which means the output values are all greater than or equal to zero. Thus, the range of f(x) = x^2 is y ≥ 0.
Which of the following sets can be the domain of a function defined as f(x) = √x?
x ≥ 0
All real numbers
x < 0
x > 0
The square root function is defined only for nonnegative inputs, meaning the domain must include zero and all positive numbers. Therefore, the correct domain is x ≥ 0.
For the function f(x) = 1/(x-2), what is the domain?
All real numbers except x = 2
x ≠ 0
x > 2
x ≠ 1
The expression 1/(x-2) is undefined when the denominator is zero, which happens when x = 2. Thus, the domain is all real numbers except for x = 2.
If a function f is defined by the set of ordered pairs {(1,2), (2,3), (3,2)}, which statement is true?
f is a function only if inputs are consecutive integers
f is not defined due to duplicate y-values
f is not a function because outputs must be unique
f is a function despite having repeated outputs
A relation is a function if each input corresponds to one output, regardless of whether the outputs repeat. In this set, every input is unique, so f qualifies as a function.
Consider the mapping diagram: {1→4, 2→4, 3→5, 4→6}. Which statement is true regarding this relation?
It is not a function because two inputs map to the same output
It is a function even if outputs repeat
It is not a function if any output is duplicated
It is a function only if each output is unique
A function is determined by the rule that every input must have exactly one output, not by the uniqueness of the outputs. Since each input in the mapping diagram has a single corresponding output, it defines a function.
For the function f(x) = |x|, which of the following statements is true?
f(x) is always positive
f(x) can produce negative outputs when x is negative
f(x) is always zero
f(x) is never negative
The absolute value function returns nonnegative results for any real number input. This means that f(x) = |x| never yields a negative value.
If f(x) = x + 3 and g(x) = 2x, what is the value of (f ∘ g)(2)?
10
7
9
8
To compute (f ∘ g)(2), first evaluate g(2) which is 2×2 = 4, then substitute into f to get f(4) = 4 + 3 = 7. Therefore, the answer is 7.
Which of the following best describes the term 'inverse function'?
A function that multiplies the original by zero
A function that is defined only on nonnegative numbers
A function that is identical to the original
A function that undoes the effect of an original function
An inverse function reverses the effect of the original function so that applying the function and its inverse returns the starting input. This concept is vital in solving equations and understanding function composition.
What is the vertical line test used for?
To establish the range of a function
To determine if a graph is symmetric
To determine the domain of a function
To confirm if a graph represents a function
The vertical line test is used to confirm whether a graph represents a function by ensuring that each vertical line intersects the graph at most once. This guarantees that each input yields a single output.
Determine the x-intercept of the function f(x) = 3x - 9.
x = 9
x = 3
x = -9
x = -3
To find the x-intercept, set f(x) to zero: 3x - 9 = 0, which simplifies to x = 3. This is the point where the graph crosses the x-axis.
Consider the function f defined by f(x) = (x^2 - 4)/(x - 2). Which statement is correct regarding f(x) when x = 2?
f(x) is undefined at x = 2
f(x) equals 4 at x = 2
f(x) equals 2 at x = 2
f(x) equals 0 at x = 2
Even though the expression simplifies to x + 2 for values other than 2, the original function is undefined at x = 2 because it results in division by zero. This creates a hole in the graph at that point.
When graphing a piecewise function, which strategy is most effective for identifying its overall domain?
Consider only the overlapping part of all pieces
Use only the domain of the first piece
Consider only the endpoints of each piece
Combine the domains of all individual pieces
The overall domain of a piecewise function is found by taking the union of the domains from each individual piece. This method ensures that all input values where the function is defined are included.
Given the function f(x) = 1/(√(x-1)), what is the domain of f?
x > 0
x > 1
x ≥ 0
x ≥ 1
The square root in the denominator requires that its argument, x - 1, be positive; moreover, the denominator cannot be zero. Therefore, x must be greater than 1 for the function to be defined.
For the composite function (f ∘ g)(x) where f(x) = √x and g(x) = x^2 - 4, which x values are in the domain of the composite function?
x such that x ≥ 4
All real x
x such that x^2 - 4 > 0
x such that x^2 - 4 ≥ 0
For the composite function (f ∘ g)(x) to be defined, the output of g(x) must be within the domain of f, which requires g(x) ≥ 0. Solving the inequality x^2 - 4 ≥ 0 yields the correct domain condition.
A function f is one-to-one if it passes which test?
Horizontal line test
Vertical line test
Two-point test
Diagonal line test
A one-to-one function produces unique outputs for unique inputs, which is verified by the horizontal line test. If any horizontal line intersects the graph more than once, the function is not one-to-one.
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Study Outcomes

  1. Apply function notation to express mathematical relationships.
  2. Analyze different types of functions to identify linear and non-linear patterns.
  3. Evaluate domains and ranges to determine viable input and output values.
  4. Interpret graphs and equations to distinguish key characteristics of functions.
  5. Assess changes in function behavior based on modifications to their equations.

2-1 Quiz: Characteristics of Functions Part 1 Cheat Sheet

  1. Functions Are Fair Deals - A function is like a perfect vending machine: every button (input) gets you exactly one snack (output), no surprises! This means you'll never press one button and get two treats. Grab a deeper dive with Symbolab's Function Overview.
  2. Function Notation Magic - When you see f(x), think of f as the machine name and x as the button you press. For example, if f(x) = x², pressing button 3 gives you 9. Learn all the neat tricks in Lumen Learning's Notation Guide.
  3. Plug-and-Play Evaluation - Evaluating a function is just feeding it numbers: if f(x) = 2x + 3, then f(5) = 2·5 + 3 = 13. It's like putting coins in a slot and watching the result. Practice more fun examples with Symbolab's Evaluation Exercises.
  4. Domain & Range Detective - The domain is your list of allowed inputs, and the range is everything the function can spit out. For √x, you can only press buttons ≥ 0, and you'll never get a negative snack! Explore hands-on tips at Symbolab's Domain & Range.
  5. Vertical Line Test Adventure - Draw a vertical line through your graph; if it crosses more than once, it's not a function. It's like checking if two candies fall from the same chute - only one candy should drop! Try interactive demos on Symbolab's Vertical Line Test.
  6. One-to-One Heroics - One-to-one means each snack is uniquely tied to one button. Use the horizontal line test: if a horizontal swipe hits the curve twice, you've got duplicates! Master inverses with Symbolab's One-to-One Guide.
  7. Meet the Function Family - From constant (f(x)=c) and identity (f(x)=x) to absolute value, quadratic, cubic, reciprocal, and root functions - each has its own personality and graph shape. Knowing their looks makes spotting them in the wild a breeze! Check out examples in Symbolab's Function Types.
  8. Is It a Function? - Test any set of pairs by making sure no input repeats with different outputs. {(1,2),(2,3),(3,4)} is golden, but {(1,2),(1,3),(2,4)} fails the vibe check! Practice more with Symbolab's Relation Checker.
  9. Graphing Your Function Journey - Every point (x, f(x)) is a pin on your chalkboard map. Plotting these points reveals peaks, valleys, and the overall personality of your function. Visual learners, rejoice with Symbolab's Graphing Guide.
  10. Range-Finding Expedition - Analyze your formula or graph to see how far outputs can roam. For f(x)=x², outputs are always ≥ 0 since squares don't go negative. Unlock tips for tricky ranges at Symbolab's Range Strategies.
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