Functions as a special kind of relation

Correct answer: A. It is a function from (A) to (B)

Each element of (A) has exactly one image in (B). In exams, repetition of the second component does not make a function invalid.

Correct answer: A. Because (1) has two different images

In a function, one domain element cannot have two different images. In exams, check repeated first components carefully.

Correct answer: A. Because (6) has no image

For a function, every element of (A) must have exactly one image. In exams, identify a missing domain element quickly.

Correct answer: B. (9)

Each of the (2) elements of (A) has (3) choices in (B), so total functions are (3^2=9). In exams, remember the formula (n(B)^{n(A)}).

Correct answer: C. (8)

Here (n(A)=3) and (n(B)=2), so total functions are (2^3=8). In exams, the base is the number of elements in the codomain.

Correct answer: C. ({2,4,6})

Putting (x=1,2,3) gives images (2,4,6). In exams, the range is only the set of actually obtained images.

Correct answer: B. 1

The number 3 is odd. According to the given rule, \(f(x)=1\) for every odd \(x\); therefore, \(f(3)=1\). The value 0 applies only when the input is even. In an exam, first check whether the input is even or odd, and then apply the definition.

Correct answer: A. It is a function and each image is (1) more

Every first component has exactly one image and it is (x+1). In exams, identifying the rule from ordered pairs is useful.

Correct answer: C. Some (a\in A) has two different images

In a function, one domain element cannot be associated with two different images. This is the most common exam mistake.

Correct answer: A. ({(1,a),(2,b),(3,c)})

In option (A), every element (1,2,3) of (A) has exactly one image. In exams, match the list of first components with (A).

Correct answer: B. 7

Substituting \(x=5\) into the given rule gives \(f(5)=5+2=7\). Hence, the correct value is 7. The value 5 is only the input, not the output of the function. Exam tip: To find a function value, substitute the given input directly into the function rule.

Correct answer: B. ({1,2,5})

(g(0)=1), (g(1)=2), and (g(2)=5), so the range is ({1,2,5}). In exams, apply the rule to every domain element.

Correct answer: A. It is a function

For each (x), (y=x^2) gives only one image. Two different (x) values having the same image does not invalidate a function.

Correct answer: B. 2

For a function, every element of the domain \(A\) must have exactly one image. Here, the first component \(2\) appears in \((2,6)\) and \((2,7)\), so it has two different images and violates the function condition. In contrast, \(1\) and \(3\) each have exactly one image. Exam tip: check the first components of the ordered pairs; a repeated first component with different second components indicates a violation.

Correct answer: B. ({1,2})

The range is the set of distinct obtained images, so it is ({1,2}). In exams, write repeated images only once.

Correct answer: A. (4^2=16)

There are (4) choices for each of the (2) elements of (A), so there are (4^2=16) functions. In exams, the exponent is the number of domain elements.

Correct answer: B. (2)

In a constant function, all elements have the same image, chosen from one element of (B). Therefore there are (2) constant functions.

Correct answer: C. Range

The set of actually obtained images is called the range. In exams, the range is always a subset of the codomain.

Correct answer: D. (5)

The range is ({2,3,4}), so (5) is in the codomain but not in the range. In exams, keep codomain and range separate.

Correct answer: D. ({(1,a),(2,b),(3,c),(3,d)})

In option (D), (3) has two images (c) and (d), and (4) has no image. In exams, check both conditions.

Correct answer: B. (y=2x)

In every ordered pair, the second component is (2) times the first. In exams, test the rule on two pairs and confirm with the rest.

Correct answer: B. 64

Since \\(f(x)=x^3\\), we substitute 4 for x: \\(f(4)=4^3=4\times4\times4=64\\). Therefore, option B is correct. The value 16 is \\(4^2\\), not the required cube. Exam tip: apply the function rule directly to the given input.

Correct answer: A. (2^6=64)

(A\times B) has (3\times 2=6) elements, so the number of relations is (2^6=64). In exams, counting relations and functions is different.

Correct answer: B. (2^3=8)

The number of functions is (n(B)^{n(A)}=2^3=8). In exams, do not confuse it with the number of relations (2^{n(A)n(B)}).

Correct answer: B. (f) is a function because every (x) has one definite image

For every real (x), (x^2) gives one definite real number. Having the same image does not break the function condition.