Relations and F...

If \(f(x)=x^{4}-\frac{1}{x^{4}}\), then \(f(x)+f\left(\frac{1}{x}\right)=?\)

The 2 functions in \(x\) are \(f(x)=e^{2 x}\) and \(g(x)=\ln x\), then find \(\operatorname{gof}(x)\)?

In the set of real numbers \(\mathrm{R}\), an operation * is defined by \(\mathrm{a} * \mathrm{b}=\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}\). Then, value of \(\left(3 {*} 4\right) \times 5\):

Let, \(R=\{(a, b): a, b \in Z\) and \((a+b)\) is even \(\}\), then \(R\) is:

If \(f(x)=\frac{x+1}{x-1}, x \neq 1\), then \(f\{f(x)\}=?\)

If \({ }^{*}\) is a binary operation on \(\mathrm{Z}\) such that \(\mathrm{a}^{*} \mathrm{~b}=\mathrm{a}+\mathrm{b}+1 \forall \mathrm{a}, \mathrm{b} \in \mathrm{Z}\) then find the identity element of \(\mathrm{Z}\) with respect to \({ }^{\star}\)?

If the function is onto and one-to-one, then it is called as ________.

If \(f: R \rightarrow R\) and \(g: R \rightarrow R\) are two mappings defined as \(f(x)=3 x\) and \(g(x)=3 x^{2}+9\), then the value of \((f+g)(2)\) is:

Let \(R\) be a relation defined as \(R=\left\{(a, b): a^{2} \geq b\right.\), where \(a\) and \(b \in Z\}\). Then, relation \(R\) is a/an:

What is the inverse of the function \(y=5^{\log x}\)?

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?

Let \({f}: {R} \rightarrow {R}\) be defined by \({f}({x})=2 {x}+6\) which is a bijective mapping, then \({f}^{-1}({x})\) is given by,

Let A = Q X Q and let * be a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for all (a, b), (c, d) belongs to A then find identity element in A.

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

The identity element for the binary operation * defined on \(Q-\{0\}\) as: \(a{*} b=\frac{a b}{2}\), is:

The relation \(R=\{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)\}\) on a set \(A=\{1,2,3\}\) is:

Which of the following functions, \(f: R \rightarrow R\) is one-one?

Let \(R\) be the set of real numbers and * be the binary operation defined on \(R\) as \(a^{*} b=a+b-a b \forall a, b \in R\). Then, the identity element with respect to the binary operation * is:

The relation 'has the same father as' over the set of children is:

If \(f(x+1)=x^{2}-3 x+2\), then what is \(f(x)\) equal to?

On the set of positive rationals, a binary operation * is defined by \(\mathbf{a} * \mathrm{b}=\frac{2 \mathrm{ab}}{5}\). If \(2{*} x=3^{-1}\), then \(x=?\)