SA

Let $A = \{-1, 0, 1, 2\}$, $B = \{-4, -2, 0, 2\}$ and $f, g : A \to B$ be functions defined by $f(x) = x^2 - x, x \in A$ and $g(x) = 2 \left| x - \frac{1}{2} \right| - 1, x \in A$. Are $f$ and $g$ equal? Justify your answer.

VSA

Determine the range of $f(x) = \cos x + \sin x + \sqrt{2}$.

LA

34. (a) If N denotes the set of all natural numbers and R is the relation on $N \times N$ defined by $(a, b) R (c, d)$, if $ad(b+c)=bc(a+d)$. Show that R is an equivalence relation.

LA

Prove that a function $f : [0, \infty) \to [-5, \infty)$ defined as $f(x) = 4x^2 + 4x - 5$ is both one-one and onto.

LA

Show that a function $f : R \to R$ defined by $f(x) = \frac{2x}{1 + x^2}$ is neither one-one nor onto. Further, find set $A$ so that the given function $f : R \to A$ becomes an onto function. [All India 2023, Delhi 2024]

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