Definition of a Function: A Comprehensive Guide for Nursing Students

Question 1

Which of the following is NOT a defining characteristic of a function?

Accepted Answer

The graph of a function is necessarily linear.

Answer

Each input value in the domain corresponds to exactly one output value in the range.

Answer

The domain and range may share common elements.

Question 2

Which of the following best defines a function?

Accepted Answer

A relation that assigns a unique value from a set to each element of another set.

Answer

A set of ordered pairs where the first element of each pair is related to the second element by a specific rule.

Answer

A graph that shows the relationship between two variables.

Answer

An equation that describes the relationship between two variables.

Question 3

Which characteristic is NOT a defining feature of a function?

Accepted Answer

Input values can be repeated.

Answer

For each input, there is exactly one output.

Answer

Input and output values can be any real number.

Answer

Output values depend on input values.

Question 4

Which of the following best defines a function?

Accepted Answer

A relationship where each input has exactly one output.

Answer

A mapping that preserves the order of elements.

Answer

An equation that shows the connection between variables.

Answer

A set of pairs with unique first elements.

Question 5

What is the precise definition of a function?

Accepted Answer

A relation that assigns to each element in a domain exactly one element in a range.

Answer

A relation that may assign to each element in a domain one or more elements in a range.

Answer

A set of ordered pairs where each first element is paired with multiple second elements.

Question 6

The defining characteristic of a function is that:

Accepted Answer

Each input has exactly one output.

Answer

The input and output values are always positive.

Answer

The graph of the function is a straight line.

Answer

The function can be represented by an equation.

Question 7

For the function f(x) = √(x - 3), what is the smallest possible value of x in its domain?

Accepted Answer

3

Answer

0

Answer

-3

Answer

Any real number

Question 8

Which graph represents the function f(x) = 2x + 3?

Accepted Answer

A line with a slope of 2 and y-intercept of 3.

Answer

A parabola opening upwards.

Answer

A circle centered at the origin.

Answer

A hyperbola with asymptotes y = 2x and y = -2x.

Question 9

The equation of the vertical line passing through the point (4, -1) is:

Accepted Answer

x = 4

Answer

y = -1

Answer

f(x) = -1

Answer

f(x) = 4

Question 10

A relation is considered a function if:

Accepted Answer

Each input corresponds to exactly one output.

Answer

The relation can be represented by a graph.

Answer

The relation is defined for all real numbers.

Answer

The relation is symmetric.

Question 11

To shift the graph of f(x) = x^2 to the left by 2 units, what transformation should be applied?

Accepted Answer

f(x - 2)

Answer

f(x + 2)

Answer

f(2x)

Answer

f(x) - 2

Question 12

A composite function of f(x) = x^2 and g(x) = x - 1 can be expressed as:

Accepted Answer

f(g(x))

Answer

g(f(x))

Answer

f(x) + g(x)

Answer

f(x) - g(x)

Question 13

Which of the following applications of functions is less relevant to the field of nursing?

Accepted Answer

Solving complex differential equations

Answer

Calculating drug dosages based on patient weight

Answer

Predicting blood pressure based on age

Answer

Analyzing the spread of infectious diseases

Question 14

Which function notation shows the relationship: 'The output is 3 more than the input'?

Accepted Answer

f(x) = x + 3

Answer

f(x) = 3x + 1

Answer

f(x) = x - 3

Question 15

Determine if the relation {(1, 2), (3, 4), (1, 5), (2, 6)} represents a function.

Accepted Answer

No, because an input value corresponds to more than one output value.

Answer

Yes, because each input value corresponds to at least one output value.

Question 16

For the function f(x) = 3x - 2, what happens to the output when the input is increased by 1?

Accepted Answer

Increases by 3

Answer

Increases by 1

Answer

Decreases by 1

Answer

Decreases by 3

Question 17

Consider the function f(x) = 2x + 3. What is the domain of this function?

Accepted Answer

All real numbers

Answer

Only positive real numbers

Answer

Only negative real numbers

Question 18

For the function f(x) = 2x - 3, what is the value when x equals -1?

Accepted Answer

-5

Answer

1

Answer

5

Answer

-15

Question 19

Determine the inverse function of f(x) = 2x + 3.

Accepted Answer

f^-1(x) = (x - 3) / 2

Answer

f^-1(x) = 2x - 3

Answer

f^-1(x) = 3 - 2x

Answer

f^-1(x) = x / 2 + 3

Question 20

The formal definition of a function is:

Accepted Answer

A relation that assigns to each element of a set a unique element of another set.

Answer

A set of ordered pairs where the second element is unique for each first element.

Answer

A mapping from one set to another set without any restrictions.

Question 21

Which of the following is not a defining characteristic of a function?

Accepted Answer

For each input, there can be multiple outputs.

Answer

There exists a range.

Answer

It can be expressed using function notation.

Answer

There exists a domain.

Question 22

Find the domain of the function f(x) = √(x - 2).

Accepted Answer

[2, ∞)

Answer

[0, ∞)

Answer

(-∞, ∞)

Answer

[-2, ∞)

Question 23

Provide the accurate definition of a function.

Accepted Answer

A relation that assigns to each element of the domain exactly one element of the range

Answer

A mapping from one set to another where each element in the first set may have multiple corresponding elements in the second set

Answer

A collection of ordered pairs where each first element occurs only once

Question 24

Determine if the relation {(1, 2), (2, 3), (3, 4)} qualifies as a function. Explain your reasoning.

Accepted Answer

Yes, it is a function because each element in the domain (1, 2, 3) has exactly one corresponding element in the range (2, 3, 4).

Answer

No, it is not a function because the relation does not pass the vertical line test

Answer

Cannot be determined from the given information

Answer

No, it is not a function because the relation is not defined for all real numbers.

Question 25

Which graph accurately represents a function?

Accepted Answer

Graph with a vertical line passing through each point

Answer

Graph with a horizontal line passing through each point

Answer

Graph with a diagonal line passing through each point

Question 26

Which of the following functions is one-to-one?

Accepted Answer

f(x) = x^3

Answer

f(x) = |x|

Answer

f(x) = x^2

Question 27

Identify the property of functions that is incorrect.

Accepted Answer

Every element in the domain must have multiple corresponding elements in the range

Answer

The domain and range of a function can be any set

Answer

Functions can be represented graphically, algebraically, or verbally

Question 28

**Define a function.**

Accepted Answer

A relation that assigns to each element of a set a **unique** element of another set

Answer

A set of all possible input values

Answer

A collection of ordered pairs

Question 29

What is the **domain** of a function?

Accepted Answer

The set of all **possible input** values

Answer

The set of all possible output values

Answer

The graph of the function

Answer

The range of the function

Question 30

What is the **range** of a function?

Accepted Answer

The set of all **possible output** values

Answer

The graph of the function

Answer

The domain of the function

Answer

The set of all possible input values

Question 31

Which of the following is **not** a property of a function?

Accepted Answer

Every input has **multiple** outputs

Answer

The function is a relation

Answer

The domain and range are sets

Answer

The function can be represented graphically

Question 32

Consider the relation {(1, 2), (2, 4), (3, 6)}. Is this relation a function?

Accepted Answer

Yes

Answer

Only if the domain is restricted to {1, 2, 3}

Answer

Cannot be determined

Answer

No

Question 33

Which of the following graphs represents a function?

Accepted Answer

A graph that passes the **vertical line** test

Answer

A graph that is a horizontal line

Answer

A graph that is a circle

Answer

A graph that is a parabola

Question 34

What is the **inverse** of a function?

Accepted Answer

A function that **undoes** the original function

Answer

A function that has the same domain and range as the original function

Answer

A function that is the reciprocal of the original function

Answer

A function that is the reflection of the original function over the line y = x

Question 35

What is the composition of the functions f(x) = x^2 and g(x) = x + 1?

Accepted Answer

f(g(x)) = x^2 + 2x + 1

Answer

f(g(x)) = x^2 + 1

Answer

f(g(x)) = x + 2

Answer

f(g(x)) = x^4 + x^2 + 1

Question 36

Which of the following graphs represents a function?

Accepted Answer

A horizontal line

Answer

A parabola

Answer

A circle

Question 37

What does the vertical line test determine?

Accepted Answer

Whether a relation is a function

Answer

Whether a function has an inverse

Answer

Whether a graph is continuous

Question 38

Which of the following is a unique characteristic of functions?

Accepted Answer

For each input, there is exactly one output

Answer

They are always increasing

Answer

They have a constant rate of change

Question 39

What is the formal definition of a function?

Accepted Answer

A relation that assigns to each element of a nonempty set a unique element of another set.

Answer

A set of ordered pairs without any repeated pairs.

Answer

A graph that depicts the relationship between a dependent variable and an independent variable.

Question 40

In the function f(x) = 2x + 1, what is the domain?

Accepted Answer

The set of all real numbers.

Answer

The set of positive real numbers.

Answer

The set of whole numbers.

Answer

The set of negative real numbers.

Question 41

Which of the following is an example of a vertical line test?

Accepted Answer

Drawing a vertical line parallel to the y-axis and checking if it intersects the graph of the relation at most once for all points on the line.

Answer

Drawing a horizontal line parallel to the x-axis and checking if it intersects the graph of the relation at most once for all points on the line.

Question 42

What is the inverse of the function f(x) = 2x + 1?

Accepted Answer

f^-1(x) = (x - 1)/2

Answer

f^-1(x) = 2x - 1

Answer

f^-1(x) = 1 - 2x

Answer

f^-1(x) does not exist.

Question 43

Which of the following is a composition of functions?

Accepted Answer

f(g(x))

Answer

f(x) - g(x)

Answer

f(x) + g(x)

Answer

f(x) * g(x)

Question 44

What is the difference between a relation and a function?

Accepted Answer

A function is a relation where each input has exactly one output, while a relation does not necessarily have this property.

Answer

A function is a relation that is graphed on a line, while a relation is graphed on a curve.

Question 45

Which of the following functions is linear?

Accepted Answer

f(x) = 2x - 3

Answer

f(x) = sin(x)

Answer

f(x) = x^2

Answer

f(x) = |x|

Question 46

Which of the following accurately defines a function?

Accepted Answer

A relation that assigns a unique element from one set to each element of another set

Answer

A rule that assigns a value to a variable

Answer

A collection of ordered pairs representing a mathematical relationship

Question 47

Identify the domain of the function f(x) = x^2 + 1:

Accepted Answer

All real numbers

Answer

Only negative real numbers

Answer

Only positive real numbers

Answer

Only the set of integers

Question 48

What is the range of the function f(x) = 1/x?

Accepted Answer

All real numbers except zero

Answer

All negative real numbers

Answer

[0, 1]

Answer

All positive real numbers

Question 49

Which of these relations is not a function?

Accepted Answer

{(1, 2), (1, 3), (2, 4)}

Answer

{(1, 2), (3, 4), (5, 6)}

Answer

{(1, 1), (2, 2), (3, 3)}

Question 50

Which function is linear?

Accepted Answer

f(x) = 2x + 1

Answer

f(x) = 2^x + 1

Answer

f(x) = sin(x) + 1

Answer

f(x) = x^2 + 1

Question 51

Find the inverse of the function f(x) = x^2:

Accepted Answer

f^-1(x) = ±√x

Answer

f^-1(x) = x^3

Answer

f^-1(x) = 1/x^2

Answer

f^-1(x) = x + 1

Question 52

Which operation does not preserve functions?

Accepted Answer

Composition with a non-function

Answer

Multiplication

Answer

Subtraction

Answer

Addition

Question 53

What's the key difference between a function and a relation?

Accepted Answer

A function is a relation where each input corresponds to only one output.

Answer

A relation is a function that is graphed on a coordinate plane.

Answer

A function is a relation that is represented by an equation.