LA

Show that the function $f : R \to \{x \in R : -1 < x < 1\}$ defined by $f(x) = \frac{x}{1 + |x|}, x \in R$ is one-one and onto function.

LA

Show that the relation $R$ in the set $A = \{1, 2, 3, 4, 5\}$ given by $R = \{(a, b) \in A : |a - b| \text{ is even}\}$, is an equivalence relation. Show that all the elements of $\{1, 3, 5\}$ are related to each other and all the elements of $\{2, 4\}$ are related to each other, but no element of $\{1, 3, 5\}$ is related to any element of $\{2, 4\}$.

MCQ_SINGLE

A function \(f:R_{+}\rightarrow R\) (where \(R_{+}\) is the set of all non-negative real numbers) defined by \(f(x)=4x+3\) is:

LA

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

LA

Let $f : \mathbb{R} - \left\{ \frac{4}{3} \right\} \to \mathbb{R}$ be a function defined as:$$f(x) = \frac{4x}{3x+4}$$Show that $f$ is a one-one function. Also, check whether $f$ is an onto function or not.

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