Relations and Functions MCQ

Answer: d) Non-linear

Explanation: In mathematics, a relation is a set of ordered pairs that establishes a relationship between elements of two sets. There are different types of relations such as reflexive, symmetric, and transitive relations.

Non-linear is not a type of relation. Non-linear typically refers to functions that do not have a straight line as their graph. 

Answer: b) {(1, 1), (2, 1), (1, 2)}

Explanation: A relation R on a set A is reflexive if for every element a in A

it is true that (a, a) ∈ R. In other words, every element in A is related to itself.

b) {(1, 1), (2, 1), (1, 2)} is an example of a reflexive relation since (1, 1), (2, 2) and (3, 3) are all elements of the set and (1, 1), (2, 2) and (3, 3) are all related to themselves.

Answer: a) f(x) = x + 1

Explanation: An onto function (also called a surjective function) is a function where every element in the range is the image of at least one element in the domain.

a) f(x) = x + 1 is an example of an onto function since for every y in the range, there exists an x in the domain such that f(x) = y.

Answer: a) {(1, 2), (2, 1), (3, 1)}

Explanation: A relation R on a set A is symmetric if for every pair of elements (a, b) in A, if (a, b) ∈ R then (b, a) ∈ R. In other words, if a is related to b, then b is related to a.

a) {(1, 2), (2, 1), (3, 1)} is an example of a symmetric relation since (1, 2) implies (2, 1) and (2, 1) implies (1, 2).

Answer:

c) f(x) = |x|

Explanation: A function f is one-to-one (also called injective) if each element in the range corresponds to exactly one element in the domain.

c) f(x) = |x| is an example of a one-to-one function since for every y in the range, there exists exactly one x in the domain such that f(x) = y.

Answer: a) {(1, 2), (2, 3), (1, 3)}

Explanation: A relation R on a set A is transitive if for every a, b, and c in A, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.

a) {(1, 2), (2, 3), (1, 3)} is an example of a transitive relation since (1, 2) and (2, 3) implies (1, 3).

Answer: b) f(x) = x²

Explanation: A function that is neither onto nor one-to-one maps some elements in the domain to the same element in the range, and/or has some elements in the range that are not mapped to by any element in the domain.

b) f(x) = x² is an example of a function that is neither onto nor one-to-one since every positive number in the range has two preimages in the domain (one positive and one negative).

Answer: a) f(x) = x², g(x) = x + 1

Explanation: A composite function (or function composition) is formed by applying one function to the result of another function.

a) f(x) = x² and g(x) = x + 1 are functions, but they are not composed with each other.

The composite function that is formed by applying g to f is denoted as g(f(x)) and can be expressed as g(f(x)) = (x² + 1).

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

Explanation: The inverse function of f(x) is denoted as f^-1(x) and is defined as the function that undoes the action of f(x), i.e., f^-1(f(x)) = x.

To find the inverse of f(x) = 2x - 3,

Replace f(x) with y: y = 2x - 3

Solve for x in terms of y: x = (y + 3)/2

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

Therefore, the inverse of f(x) = 2x - 3 is f^-1(x) = (x + 3)/2, which is option (b).

Answer: b) f(x) = e^x

Explanation: A function that is both onto and one-to-one maps every element in the domain to a unique element in the range, and maps every element in the range to by some element in the domain.

b) f(x) = e^x is an example of a function that is both onto and one-to-one since every positive number in the range has a unique preimage in the domain, and every element in the domain has a unique image in the range.

Answer: b) [-3, 3]

Explanation: The domain of a function is the set of all possible input values (x) for which the function is defined.

For f(x) = sqrt(9 - x²), the radicand (9 - x²) cannot be negative, so we have the inequality 9 - x² >= 0. Solving for x, we get -3 <= x <= 3.

Therefore, the domain of f(x) is [-3, 3], which is option (b).

Answer: d) Test whether every element in the range has a preimage in the domain.

Explanation: A function is onto if every element in the range is mapped to by at least one element in the domain.

d) Testing whether every element in the range has a preimage in the domain is a way to test whether a function is onto. This means that for every element y in the range, there exists at least one element x in the domain such that f(x) = y.

Answer: a) f^-1(x) = 2^x - 1

Explanation: To find the inverse of f(x) = log(base 2)(x + 1),

Replace f(x) with y: y = log(base 2)(x + 1)

Rewrite the equation in exponential form: 2^y = x + 1

Solve for x in terms of y: x = 2^y - 1

Replace x with f^-1(x): f^-1(x) = 2^x - 1

Therefore, the inverse of f(x) is f^-1(x) = 2^x - 1, which is option (a).

Answer: a) f^-1(x) = (x + 4)^(1/3)

Explanation: To find the inverse of f(x) = x³ - 4,

Replace f(x) with y: y = x³ - 4

Solve for x in terms of y: x = (y + 4)^(1/3)

Replace x with f^-1(x): f^-1(x) = (x + 4)^(1/3)

Therefore, the inverse of f(x) is f^-1(x) = (x + 4)^(1/3), which is option (a).

Answer: c) (-∞, 1) U (2, ∞)

Explanation: To find the domain of f(x) = ln(x² - 3x + 2), we need to find the values of x for which the argument of the natural logarithm is positive. Therefore, we need to solve the inequality x² - 3x + 2 > 0.

Factorizing the quadratic, we get (x - 1)(x - 2) > 0.

The sign of the expression depends on the intervals between the roots of the quadratic, which are x = 1 and x = 2.

Therefore, the inequality is satisfied in the intervals (-∞, 1) U (2, ∞). This means that the domain of f(x) is the same as the intervals where the inequality is satisfied:

Domain = (-∞, 1) U (2, ∞)

Therefore, the correct answer is option (c).

Answer: d) x ≥ 3

Explanation: The square root function is defined only for non-negative values. Therefore, we need to find the values of x that make the expression inside the square root non-negative:

x - 3 ≥ 0

x ≥ 3

Therefore, the domain of the function f(x) = sqrt(x - 3) is x ≥ 3, which means that the correct answer is option (d).

Answer: a) x ≠ -2

Explanation: The denominator of the fraction cannot be equal to zero, which means that x + 2 ≠ 0:

x ≠ -2

Therefore, the domain of the function f(x) = 1 / (x + 2) is x ≠ -2, which means that the correct answer is option (a).

Answer: c) x > 4

Explanation: The logarithmic function is defined only for positive values. Therefore, we need to find the values of x that make the expression inside the logarithm positive:

x - 4 > 0

x > 4

Therefore, the domain of the function f(x) = log(x - 4) is x > 4, which means that the correct answer is option (c).

Answer: b) -4 ≤ x ≤ 4 

Explanation: The square root function is defined only for non-negative values. Therefore, we need to find the values of x that make the expression inside the square root non-negative:

4 - x² ≥ 0

x² ≤ 4

-2 ≤ x ≤ 2

Therefore, the domain of the function f(x) = sqrt(4 - x²) is -2 ≤ x ≤ 2, which means that the correct answer is option (b).

Answer: d) all real numbers except x = ±2

Explanation: The denominator of the fraction cannot be equal to zero, which means that x² - 4 ≠ 0:

Therefore, the domain of the function f(x) = 1 / (x² - 4) is all real numbers except x = ±2, which means that the correct answer is option (d).

Answer: a) x ≠ 4 and x > 1/2

Explanation: We need to find the values of x that make both the numerator and denominator of the function defined. The numerator is defined only for non-negative values of the expression inside the square root:

2x - 1 ≥ 0

x ≥ 1/2

The denominator is defined only if x - 4 ≠ 0:

x ≠ 4

Therefore, the domain of the function f(x) = sqrt(2x - 1) / (x - 4) is x ≠ 4 and x ≥ 1/2, which means that the correct answer is option (a).

Answer: a) x ≠ 1 and x ≠ 2

Explanation: We need to find the values of x that make both the numerator and denominator of the function defined. The numerator is defined only for positive values of the expression inside the logarithm:

2x + 3 > 0

x > -3/2

The denominator is defined only if x² - 4x + 3 ≠ 0:

(x - 1)(x - 3) ≠ 0

x ≠ 1 and x ≠ 3

Therefore, the domain of the function f(x) = log(2x + 3) / (x² - 4x + 3) is x ≠ 1 and x ≠ 3, which means that the correct answer is option (a).

Answer: d) all real numbers greater than or equal to -3

Explanation: To find the range of the function f(x) = x² - 3, we need to find the set of all possible values of f(x) for all values of x in the domain of the function. Since the domain of f(x) is all real numbers, we can consider all possible values of x.

The minimum value of x² is 0, when x = 0. Therefore, the minimum value of f(x) is -3. However, for any positive value of x, the value of f(x) will be greater than or equal to -3. For any negative value of x, the value of f(x) will be greater than or equal to -3, because the square of a negative number is positive.

Therefore, the range of the function f(x) = x² - 3 is all real numbers greater than or equal to -3, which means that the correct answer is option (d).

Answer: a) all real numbers

Explanation: To find the range of the function f(x) = 2x + 1, we need to find the set of all possible values of f(x) for all values of x in the domain of the function. Since the domain of f(x) is all real numbers, we can consider all possible values of x.

The function f(x) is a linear function, which means that its graph is a straight line with a slope of 2 and a y-intercept of 1. Therefore, the range of f(x) is all real numbers, because the line can take any y-value.

Therefore, the range of the function f(x) = 2x + 1 is all real numbers, which means that the correct answer is option (a).

Answer: a) all real numbers except 0

Explanation: To find the range of the function f(x) = 1 / (x - 3), we need to find the set of all possible values of f(x) for all values of x in the domain of the function. The domain of the function is all real numbers except x = 3, because the denominator cannot be equal to zero.

If x approaches 3 from the left side, then the value of the function approaches negative infinity. If x approaches 3 from the right side, then the value of the function approaches positive infinity. Therefore, the range of the function is all real numbers except 0, because the function cannot take the value of zero.

Therefore, the range of the function f(x) = 1 / (x - 3) is all real numbers except 0, which means that the correct answer is option (a).

Answer: a) 9x² - 6x + 1

Explanation: To find the composite function (f o g)(x), we first need to find g(x) and substitute it into f(x).

g(x) = 3x - 1

(f o g)(x) = f(g(x)) = f(3x - 1) = (3x - 1)²

Expanding the square gives:

(f o g)(x) = 9x² - 6x + 1

Therefore, the correct answer is option (a).

Answer: a) 4x² + 4x + 1

Explanation: To find the composite function(f o g)(x), we first need to find f(x) and substitute it into g(x).

f(x) = 2x + 1

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

Expanding the square gives:

(f o g)(x) = 4x² + 4x + 1

Therefore, the correct answer is option (a).

Answer: a) x³ - 1

Explanation: To find the composite function (f o g)(x), we first need to find g(x) and substitute it into f(x).

g(x) = x³

(f o g)(x) = f(g(x)) = f(x³) = x³ - 1

Therefore, the correct answer is option (a).