Functions: Definition, Types, and Transformations - Year 13 Mathematics

Question 1

A function is a relation where each element of the:

Accepted Answer

Domain is mapped to exactly one element of the range.

Answer

Range is mapped to exactly one element of the domain.

Answer

Domain is mapped to one or more elements of the range.

Answer

Range is mapped to one or more elements of the domain.

Question 2

Which graph represents the function f(x) = x² - 1?

Accepted Answer

A parabola opening upwards, shifted down by 1 unit

Answer

A parabola opening downwards, shifted down by 1 unit

Answer

A straight line with a positive slope

Answer

A straight line with a negative slope

Question 3

The graph of f(x) = x² is shifted 3 units to the right. The equation of the transformed function is:

Accepted Answer

f(x + 3)

Answer

f(x - 3)

Answer

f(x - 3) + 3

Answer

f(x + 3) + 3

Question 4

The inverse function of f(x) = 3x - 2 is:

Accepted Answer

y = (x + 2)/3

Answer

y = 3x + 2

Answer

y = x/3 + 2

Answer

y = 3x - 2

Question 5

Which graph represents the function f(x) = |x + 1|?

Accepted Answer

A V-shaped graph with vertex at (-1, 0)

Answer

A U-shaped graph with vertex at (-1, 0)

Answer

A straight line with a positive slope

Answer

A straight line with a negative slope

Question 6

The composition of the functions f(x) = x³ and g(x) = x + 2 is:

Accepted Answer

f(g(x)) = x³ + 8

Answer

g(f(x)) = x³ + 8

Answer

f(g(x)) = (x + 2)³

Answer

g(f(x)) = (x + 2)³

Question 7

An example of a piecewise function is:

Accepted Answer

f(x) = { x² if x ≥ 0, -x² if x < 0 }

Answer

f(x) = x³

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Question 8

Which characteristic is NOT true for a function?

Accepted Answer

A function can have multiple inputs for the same output.

Answer

A function assigns each input to exactly one output.

Answer

A function can be represented by an equation.

Question 9

For the function f(x) = 1/x, what is its domain?

Accepted Answer

All real numbers except 0

Answer

x greater than 0

Answer

x less than 0

Answer

All real numbers

Question 10

Which function is an example of an exponential function?

Accepted Answer

f(x) = 2^x

Answer

f(x) = 2x + 1

Answer

f(x) = x^2 - 1

Answer

f(x) = sin(x)

Question 11

What is the inverse function of f(x) = log(x)?

Accepted Answer

f^-1(x) = e^x

Answer

f^-1(x) = 10^x

Answer

f^-1(x) = 2^x

Answer

f^-1(x) does not exist

Question 12

What type of graph does the function f(x) = x² - 4 represent?

Accepted Answer

Parabola opening upwards

Answer

Parabola opening downwards

Answer

Straight line

Answer

Circle

Question 13

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

Accepted Answer

All real numbers except 0

Answer

x > 0

Answer

x < 0

Answer

x ≥ 1

Question 14

Which of the following correctly classifies the function f(x) = 2x + 3?

Accepted Answer

Linear

Answer

Quadratic

Answer

Exponential

Answer

Logarithmic

Question 15

Which transformation would shift the graph of f(x) = x² to the left by 2 units?

Accepted Answer

f(x - 2)

Answer

f(x + 2)

Answer

f(2x)

Answer

f(x / 2)

Question 16

Which of the following best describes the graph of the function f(x) = e^x?

Accepted Answer

An exponential curve

Answer

A straight line

Answer

A parabola

Answer

A circle

Question 17

What is the derivative of the function f(x) = sin(x)?

Accepted Answer

cos(x)

Answer

sin(x)

Answer

tan(x)

Answer

cot(x)

Question 18

Which equation represents the function obtained by translating the graph of f(x) = x³ three units to the right and then reflecting it over the y-axis?

Accepted Answer

g(x) = - (x - 3)³

Answer

g(x) = (x + 3)³

Answer

g(x) = (x - 3)³

Answer

g(x) = - (x + 3)³

Question 19

What is the domain of the function f(x) = √(x - 2)?

Accepted Answer

x ≥ 2

Answer

x ≤ 2

Answer

x < 0

Answer

x > 0

Question 20

Find the domain and range of the function f(x) = 1/(x - 2).

Accepted Answer

Domain: x ≠ 2, Range: y ≠ 0

Answer

Domain: x ∈ R, Range: y ≠ 0

Answer

Domain: x ≠ 0, Range: y ∈ R

Answer

Domain: x ∈ R, Range: y ≠ 2

Question 21

Classify the function f(x) = (x² - 9)/(x + 3) as either a polynomial or rational function. If it's rational, specify any restrictions on its domain.

Accepted Answer

Rational function, domain x ≠ -3

Answer

Polynomial function, no domain restrictions

Answer

Polynomial function, domain x ≠ 3

Answer

Rational function, domain x ≠ 0

Question 22

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

Accepted Answer

The graph of a function can intersect itself at multiple points

Answer

Each input corresponds to exactly one output

Answer

The set of all possible inputs is known as the domain

Answer

The set of all possible outputs is known as the codomain

Question 23

Which transformation will shift the graph of y = f(x) three units to the right?

Accepted Answer

y = f(x + 3)

Answer

y = f(x - 3)

Answer

y = f(3x)

Answer

y = 3f(x)

Question 24

Which of the following functions has a vertical asymptote at x = -1?

Accepted Answer

y = 1/(x + 1)

Answer

y = 2x - 1

Answer

y = x^2 + 1

Answer

y = e^x + 1

Question 25

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

Accepted Answer

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

Answer

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

Answer

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

Answer

f^-1(x) does not exist

Question 26

Which characteristic describes logarithmic functions?

Accepted Answer

Their domain is all positive real numbers

Answer

Their range is all real numbers

Answer

Their graph is always decreasing

Answer

Their y-intercept is (1, 0)

Question 27

Identify the critical point of the function f(x) = x^4 - 2x^2 + 1.

Accepted Answer

x = ±1

Answer

x = 0

Answer

x = ±2

Answer

The function has no critical points

Question 28

The domain of the function f(x) = 1/(x-2) is:

Accepted Answer

All real numbers except for 2

Answer

All real numbers

Answer

All real numbers greater than 2

Answer

All real numbers less than 2

Question 29

The transformation that shifts the graph of the function f(x) = |x| three units to the right is:

Accepted Answer

f(x+3) = |x|

Answer

f(x-3) = |x|

Answer

f(3x) = |x|

Answer

f(x/3) = |x|

Question 30

The derivative of the function f(x) = e^x is:

Accepted Answer

f'(x) = e^x

Answer

f'(x) = x*e^x

Answer

f'(x) = 1/e^x

Answer

f'(x) = 0

Question 31

The integral of the function f(x) = 1/(x+1) is:

Accepted Answer

ln(|x+1|) + C

Answer

1/(x+1) + C

Answer

x * ln(|x+1|) + C

Answer

1/x * ln(|x+1|) + C

Question 32

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

Accepted Answer

The output values can repeat for different input values.

Answer

Each input value corresponds to exactly one output value.

Answer

The input values form the domain of the function.

Answer

The output values form the range of the function.

Question 33

Identify the exponential function among the following options:

Accepted Answer

f(x) = 3^(x + 2)

Answer

f(x) = 2x + 1

Answer

f(x) = x^2 - 3x

Answer

f(x) = log(x - 1)

Question 34

Which transformation shifts a graph 2 units to the left?

Accepted Answer

f(x - 2)

Answer

f(x) + 2

Answer

f(x + 2)

Answer

f(2x)

Question 35

Find the derivative of the function f(x) = e^(2x).

Accepted Answer

2e^(2x)

Answer

2xe^(2x)

Answer

e^(2x)

Answer

xe^(2x)

Question 36

Determine the integral of the function f(x) = 1/(x^2 + 1).

Accepted Answer

arctan(x) + C

Answer

tan(x) + C

Answer

1/x + C

Answer

x + C

Question 37

Which of the following does NOT represent a function?

Accepted Answer

A set of ordered pairs where the first element of each pair is repeated.

Answer

A table of values that assigns a unique output to each input.

Answer

A formula that relates an input to an output.

Answer

A graph where each input has only one output.

Question 38

What is the domain of the function f(x) = √(x - 2)?

Accepted Answer

x ≥ 2

Answer

All real numbers

Answer

x ≤ 2

Answer

x > 2

Question 39

The graph of the function y = x² is shifted 3 units to the right and 2 units down. What is the equation of the transformed function?

Accepted Answer

y = (x - 3)² - 2

Answer

y = (x - 3)² + 2

Answer

y = (x + 3)² - 2

Answer

y = (x + 3)² + 2

Question 40

Which of the following is an exponential function?

Accepted Answer

f(x) = 2^x

Answer

f(x) = √x

Answer

f(x) = x²

Answer

f(x) = 3x - 1

Question 41

If the graph of a function is symmetric about the y-axis, what kind of function is it?

Accepted Answer

Even function

Answer

Linear function

Answer

Quadratic function

Answer

Odd function

Question 42

On which interval is the function f(x) = log₂(x) increasing?

Accepted Answer

x > 0

Answer

x < 0

Answer

All real numbers

Answer

x ≥ 0

Question 43

Which function has a vertical asymptote at x = 2?

Accepted Answer

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

Answer

f(x) = |x - 2|

Answer

f(x) = x² - 4

Answer

f(x) = √(x - 2)