Types of relations

Correct answer: A. (y,x) belongs to R

Direct answer: A, (y,x) belongs to R. The inverse relation is formed by reversing the order of every ordered pair. By definition, (x,y) ∈ R^-1 if and only if (y,x) ∈ R. Thus the given pair in the inverse immediately gives the reversed pair in the original relation. Option A states exactly this definition. Option B is not guaranteed because R may contain (y,x) but not (x,y). Option C is unrelated: it concerns a diagonal pair and cannot be inferred from the given information. Option D is also not necessary; symmetry would require both directions whenever one direction is present, but no such condition is given. Memory cue: inverse means “interchange the coordinates”; symmetric means “the interchange is already also present in the same relation.”

Correct answer: A. 4

To evaluate \(f(3)\), substitute \(x=3\) into the function: \(f(3)=3k+2\). Since \(f(3)=14\), we have \(3k+2=14\), so \(3k=12\) and \(k=4\). The value 3 usually results from an arithmetic error while solving \(3k=12\); 6 and 12 do not satisfy the original equation. Exam tip: substitute the given input first, isolate the unknown, and verify the result in the original function.

Correct answer: A. 5

Substitute \(x=2\) into the given function: \(f(2)=2^2+k=4+k\). Since \(f(2)=9\), we get \(4+k=9\), so \(k=5\). Option B (4) is incorrect because it would give \(f(2)=4+4=8\), not 9. Exam tip: when a function value is given, substitute the corresponding input first and then solve for the unknown constant.

Correct answer: A. 4

Substitute x = 2 into f(x): f(2) = 2² + 2k = 4 + 2k. Since f(2) = 12, 4 + 2k = 12, so 2k = 8 and k = 4. Therefore, option A is correct.

Correct answer: A. 10

In the composite function \((h\circ g\circ f)(2)\), evaluate the functions from the inside outward. First, \(f(2)=2-1=1\); then \(g(1)=3\times1=3\); finally, \(h(3)=3^2+1=10\). Therefore, the correct answer is 10. Option B, 9, results from omitting the \(+1\) in \(h(x)\). Exam tip: in \(h\circ g\circ f\), apply \(f\) first, followed by \(g\), and then \(h\).