Free Pre-Calculus Functions Questions and Answers — Questions and Answers

Question 1: What is an even function defined as?

A function f is even if f(x) is always positive.
A function f is even if f(x) equals the independent variable, x.
A function f is even if f(−x)=f(x), for all x in the domain of f. (Correct answer)
A function f is even if f(x) represents a constant.

Correct answer: A function f is even if f(−x)=f(x), for all x in the domain of f.

An even function is characterized by its symmetry about the y-axis. Mathematically, a function f is considered even if substituting -x for x in the function's expression results in the original function, i.e., f(-x) = f(x). This property means that points (x, y) and (-x, y) both lie on the graph of the function.

Question 2: How would you define an odd function?

A function f is odd if f(x) represents a constant.
A function f is odd if f(−x)=−f(x), for all x in the domain of f. (Correct answer)
A function f is odd if f(x) equals the independent variable, x.
A function f is odd if f(x) is always positive.

Correct answer: A function f is odd if f(−x)=−f(x), for all x in the domain of f.

An odd function exhibits rotational symmetry about the origin. Its definition states that if you replace x with -x in the function, the result is the negative of the original function, expressed as f(-x) = -f(x). This means that if (x, y) is a point on the graph, then (-x, -y) is also a point on the graph.

Question 3: Which of these indicate that a function is one-to-one or not?

Vertical Line Test
Horizontal Line Test (Correct answer)
Horizontal Asymptote Test
Intersection Test

Correct answer: Horizontal Line Test

The Horizontal Line Test is used to determine if a function is one-to-one. If any horizontal line intersects the graph of a function at most once, then the function is one-to-one. This means that each output (y-value) corresponds to exactly one input (x-value).

Question 4: What test shows if a relationship is a function or not?

Intersection Test
Slope Test
Horizontal Line Test
Vertical Line Test (Correct answer)

Correct answer: Vertical Line Test

The Vertical Line Test is a graphical method used to determine if a given relation is a function. If any vertical line drawn through the graph intersects the relation at more than one point, then the relation is not a function. This is because a function must assign exactly one output (y-value) to each input (x-value).

Question 5: What does the composition of functions mean?

An operation (fog)(x) that multiplies f(x) and g(x).
An operation (fog)(x) that is equal to f (g(x)). (Correct answer)
An operation (fog)(x) that is equal to f (x) + g(x).
An operation (fog)(x) that evaluates a constant value.

Correct answer: An operation (fog)(x) that is equal to f (g(x)).

The composition of functions, denoted as (f o g)(x) or f(g(x)), is an operation where one function is applied to the result of another function. Essentially, the output of the inner function (g(x)) becomes the input for the outer function (f). This creates a new function that combines the actions of both original functions.

Question 6: What is meant by the term "periodic function"?

A function that has a variable period depending on its input.
A function that repeats itself at regular intervals. (Correct answer)
A function that has an undefined output for certain inputs.
A function that contains both even and odd terms.

Correct answer: A function that repeats itself at regular intervals.

A periodic function is a function that repeats its values in regular intervals or periods. This means that its graph exhibits a repeating pattern over a specific length along the x-axis. Trigonometric functions like sine and cosine are classic examples of periodic functions.

Question 7: How do you define a function's or relation's domain?

The set of all outputs of the function or relation.
The set of all independent variables in the function or relation.
The set of all inputs for which the function or relation is defined. (Correct answer)
The set of all even values in the function or relation.

Correct answer: The set of all inputs for which the function or relation is defined.

The domain of a function or relation refers to the complete set of all possible input values (x-values) for which the function or relation produces a defined output. In simpler terms, it's all the values that you can plug into the function without causing mathematical issues like division by zero or taking the square root of a negative number.

Question 8: Piecewise function is defined as:

A function built from pieces of different functions over different intervals. (Correct answer)
A function that is undefined for all values in its domain.
A function that returns multiple outputs for a single input.
A function that has a single rule for all values in its domain.

Correct answer: A function built from pieces of different functions over different intervals.

A piecewise function is defined by multiple sub-functions, each applying to a specific interval of the independent variable's domain. This allows the function's behavior to change across different parts of its domain, creating a graph that is composed of distinct "pieces." Each piece is a standard function, but the overall function is a combination of these pieces.

Question 9: In a function, what is the role of the dependent variable?

The constant value that remains unchanged in the function.
The independent variable that determines the output.
The output variable that depends on the input, or independent variable. (Correct answer)
The variable that creates undefined situations in the function.

Correct answer: The output variable that depends on the input, or independent variable.

In a function, the dependent variable represents the output value, typically denoted as 'y' or 'f(x)'. Its value is determined by, or "depends on," the value of the independent variable. As the input (independent variable) changes, the output (dependent variable) changes accordingly based on the function's rule.

Question 10: In a function, what is the independent variable?

The variable that creates undefined situations in the function.
The variable that does not depend on the other variable (Correct answer)
The output variable that depends on the input.
The constant value used to evaluate the function.

Correct answer: The variable that does not depend on the other variable

The independent variable, often denoted as 'x', is the input to a function whose value can be chosen freely. It is the variable that causes a change in the dependent variable. Its value is not determined by any other variable within the context of the function.

Question 11: What is a function's or relation's range?

The set of all even values in the function or relation.
The set of all inputs for which the function or relation is defined.
The set of all independent variables in the function or relation.
The set of all possible outputs of the function or relation. (Correct answer)

Correct answer: The set of all possible outputs of the function or relation.

The range of a function or relation is the collection of all possible values that the dependent variable (output) can take. It represents the set of all 'y' values that are produced when the function is evaluated for every value in its domain. Understanding the range helps to describe the full extent of a function's output.

Question 12: Undefined function is ____ ?

A function that contains both even and odd terms.
A function that involves complex numbers in its calculations.
A function that lacks a valid rule or expression for certain input values in its domain. (Correct answer)
A function that has multiple outputs for a single input.

Correct answer: A function that lacks a valid rule or expression for certain input values in its domain.

An undefined function, for specific input values, means that for those inputs, the function's rule does not yield a real number output. Common scenarios include division by zero, taking the square root of a negative number, or taking the logarithm of a non-positive number. These situations indicate that the function's expression is not valid for those particular domain values.

Question 13: What does the inverse of a function or a relationship mean?

A function or relation with negative outputs for positive inputs.
A function or relation that is equal to its original form.
A function or relation with inputs greater than its outputs.
A correspondence from the range to the domain, found by interchanging the variables. (Correct answer)

Correct answer: A correspondence from the range to the domain, found by interchanging the variables.

The inverse of a function or relation essentially reverses the mapping of the original function. It takes the output values of the original function as its inputs and produces the original input values as its outputs. Mathematically, this is achieved by swapping the roles of the independent and dependent variables (x and y) and then solving for the new dependent variable.

Question 14: Defined function is ____ ?

A function that assigns an output for a given value of its independent variable. (Correct answer)
A function that contains both even and odd terms.
A function that has multiple outputs for a single input.
A function that involves imaginary numbers in its calculations.

Correct answer: A function that assigns an output for a given value of its independent variable.

A defined function means that for every valid input value within its domain, the function produces a unique and real output value. This implies that there are no mathematical inconsistencies, like division by zero or square roots of negative numbers, for the given input. When a function is defined, it consistently maps each input to a specific output according to its rule.

Question 15: One-to-one function is defined as ___ ?

A function where each element in its domain is paired with exactly one element from its range.
A function with infinite elements in its domain.
A function that has a constant output for all inputs.
A function where each element in its range is paired with exactly one element from its domain. (Correct answer)

Correct answer: A function where each element in its range is paired with exactly one element from its domain.

A one-to-one function, also known as an injective function, ensures that every distinct input value maps to a distinct output value. In simpler terms, no two different input values will ever produce the same output value. This property is crucial for a function to have an inverse that is also a function.

Question 16: If g(x) = x^4 + 3, then g (2) is equal to?

11
14
19 (Correct answer)
18

Correct answer: 19

To evaluate g(2), substitute x = 2 into the function g(x) = x^4 + 3. This yields g(2) = (2)^4 + 3. Calculating 2^4 gives 16, so g(2) = 16 + 3, which simplifies to 19.

Free Pre-Calculus Functions Questions and Answers

What is an even function defined as?