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Relations and Functions

Multiple Choice Questions

491.

For a, b $\in$ R, define $a * b = \frac{a}{a + b}$ , where a + b $\neq$ 0. If a * b = 5, then the value of b * a is

492.

Let A = {x, y, z} and B = {a, b, c, d}. Which one of the following is not a relation from A to B ?

{(x, a), (x, c)}
{(y, c), (y, d)}
{(z, a), (z, d)}
{(z, b), (y, b), (a, d)}

493.

If f(x) = x² - 1 and g(x) = (x + 1)², then (gof) (x) is

(x + 1)⁴ - 1
x⁴ - 1
x⁴
(x + 1)⁴

494.

If the function f : $[1, \infty) \to [1, \infty)$ is defined by f(x) = 2^{x(x - 1)}, then f^-1(x) is

${(\frac{1}{2})}^{x (x - 1)}$
$\frac{1}{2} (1 - \sqrt{1 + 4 \log_{2} (x)})$
$\frac{1}{2} \sqrt{1 + 4 \log_{2} (x)}$
$\frac{1}{2} [1 + \sqrt{1 + 4 \log_{2} (x)}]$

495.

If n(A) = 8 and $n (A \cap B) = 2$ , then $n ((A \cap B)' \cap A)$ is equal to

496.
If f(x) = $\sin (x) + \cos (x)$ , $x \in (- \infty, \infty)$ and g(x) = x², $x \in (- \infty, \infty)$ , then (fog)(x) is equal to

1

0

$\sin^{2} (x) + \cos (x^{2})$

$\sin (x^{2}) + \cos (x^{2})$

$\sin (x^{2}) + \cos (x^{2})$

$f (x) = \sin (x) + \cos (x), x \in (- \infty, \infty) and g (x) = x^{2}, x \in (- \infty, \infty) fog (x) = f (g (x)) = f (x^{2}) = \sin (x^{2}) + \cos (x^{2})$

497.

If n(A) = 5 and n(B) = 7, then the number of relations on A x B is

2³⁵
2⁴⁹
2²⁵
$2^{35 \times 35}$

498.

Let $ϕ (x) = \frac{b (x - a)}{b - a} + \frac{a (x - b)}{a - b}$ , where x $\in$ R and a and b are fixed real numbers with $a \neq b$ . Then, $ϕ (a + b)$ is equal to