Relations and F...

Let \(f: R \rightarrow R\) be a function defined as \(f(x)=e^{x}\), for each \(x \in R, R\) is being the set of real numbers. Which one of the following is correct?

Consider the following statements:

Statement 1: The function \(f: R \rightarrow R\) such that \(f(x)=x^{3}\) for all \(x \in R\) is one-one.

Statement 2: \(\mathrm{f}(\mathrm{a})=\mathrm{f}(\mathrm{b}) \Rightarrow a=\mathrm{b}\) for all \(a, b \in R\) if the function \(f\) is one-one.

Which one of the following is correct in respect of the above statements?

Let \(\mathrm{L}\) denote the set of all straight lines in a plane. Let a relation \(\mathrm{R}\) be defined by \(\mathrm{lRm}\) if and only if \(\mathrm{l}\) is parallel to \(\mathrm{m}, \forall\) \(\mathrm{l,m \in L}\). Then \(\mathrm{R}\) is:

Which of the following statements is/are true:

If * is a binary operation on Z, such that:

a * b = a + b + 1 ∀ a, b ∈ Z.

1. * is associative on Z.

2. * is commutative on Z.

The relation \(\mathrm{R}\) on the set of integer is given by \(\mathrm{R}=\{(\mathrm{a}, \mathrm{b}):\) \(a - b\) is divisible by 7, where \(a, b \in Z\}\), then \(\mathrm{R}\) is a/an:

Consider the function \(f: R \rightarrow\{0,1\}\) such that:

\(f(x)=\left\{\begin{array}{c}1 \text { if } x \text { is rational } \\ 0 \text { if } x \text { is irrational }\end{array}\right.\).

Which one of the following is correct?

Which one of the following is correct?

Let \(\mathrm{N}\) be the set of natural numbers and \(\mathrm{f}: \mathrm{N} \rightarrow \mathrm{N}\) be a function given by \(\mathrm{f}(\mathrm{x})=\mathrm{x}+1 \forall \mathrm{x} \in \mathrm{N}\). Which one of the following is correct?

The function \(f(x)=x^{2}+4 x+4\) is:

If \(f(x)=8 x^{3}, g(x)=x^{\frac{1}{3}}\), then \(\operatorname{gof}(2)\) is?