Functions Questions (992)

If \(f(x)\) be a function satisfying \(f(x) \cdot f\!\left(\dfrac{1}{x}\right) = f(x) + f\!\left(\dfrac{1}{x}\right)\) and \(f(4) = 65\), then find \(f(6)\).

In a town of 10,000 families, it was found that 40% families buy newspaper A, 20% buy newspaper B and 10% buy newspaper C. Also, 5% families buy newspapers A and B, 3% buy newspapers B and C and 4% buy newspapers A and C. If 2% families buy all the three newspapers, then number of families which buy newspaper A only is

Let \(f(x) = 9^x/(9^x + 3)\) for all \(x \in \mathbb{R}\). Then the value of \(\displaystyle\sum_{r=1}^{2008} f(r/2009)\) is ______.

If \(f : A \to B\), \(f(x) = \sin^{-1}\left(\dfrac{[x]}{\{x\}}\right)\) and \(g : C \to D\), \(g(x) = \cos^{-1}\left(\dfrac{[x]}{\{x\}}\right)\), then which of the following is always correct?[Note: \([\cdot]\) and \(\{\cdot\}\) denotes greatest integer and fractional part function respectively.]

Let Z be the set of integers. If \(A = \{x \in \mathbb{Z} : 2^{(x+2)(x^2-5x+6)} = 1\}\) and \(B = \{x \in \mathbb{Z} : -3 , then the number of subsets of the set \(A \times B\) is

JM Q25.

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three non-zero non-coplanar vectors. Let the position vectors of four points A, B, C and D be $\vec{a}-\vec{b}+\vec{c}$, $\lambda\vec{a}-3\vec{b}+4\vec{c}$, $-\vec{a}+2\vec{b}-3\vec{c}$ and $2\vec{a}-4\vec{b}+6\vec{c}$ respectively. If $\overrightarrow{AB}$, $\overrightarrow{AC}$ and $\overrightarrow{AD}$ are coplanar, then $\lambda$ is:

If \(f(x) = 3x + |x|\), \(g(x) = \dfrac{3x}{4} - \dfrac{|x|}{4}\), then:

\({\sim}(p \lor q) \lor ({\sim}p \land q)\) is equivalent to:

Find the domain and range of the function \(f(x) = \sqrt{2 - x} + \sqrt{1 + x}\).

The function $f:\mathbb{R}\to\mathbb{R}$, $f(x)=\dfrac{x^2+2x-15}{x^2-4x+9}$, $x\in\mathbb{R}$ is:

A survey shows that 63% of the people in a city read newspaper A whereas 76% read newspaper B. If x% of the people read both the newspapers, then a possible value of x can be

The domain of $f(x) = \dfrac{\log_{(x+1)}(x-2)}{e^{2\log_e x} - (2x+3)}$, $x \in \mathbb{R}$ is

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

Let R1 = {(a, b) ∈N × N : |a −b| ≤13} and R2 = {(a, b) ∈N × N : |a −b| ̸= 13}. Then:

Let \(z\) be the set of integers. If \(A = \{x \in z : 2^{(x+2)(x^2 - 5x + 6)} = 1\}\) and \(B = \{x \in z : -3

If \(f: R \to R\) is defined as \(f(x) = \begin{cases} x + 4, & x

\(f(x+y)=f(x)+f(y)\), \(f(1)=2\). \(g(n)=\sum_{k=1}^{n-1}f(k)\). Find \(n\) with \(g(n)=20\).

The number of functions f from \(\{1, 2, 3, \ldots, 20\}\) onto \(\{1, 2, 3, \ldots, 20\}\) such that f(k) is a multiple of 3, whenever k is a multiple of 4, is :-

JM Q29.

For suitable \(a\), \(f(x)=\dfrac{a-x}{a+x}\) with \((f\circ f)(x)=x\). Find \(f\!\left(-\frac{1}{2}\right)\).

For some $a, b, c \in \mathbb{N}$, let $f(x) = ax - 3$ and $g(x) = x^b + c$, $x \in \mathbb{R}$. If $(f \circ g)^{-1}(x) = \left(\dfrac{x-7}{2}\right)^{1/3}$, then $(f \circ g)(ac) + (g \circ f)(b)$ is equal to ____.

If \(p\), \(q\) and \(r\) are simple propositions with truth values T, F and T, respectively, then the truth value of \((\sim p \vee q) \wedge \sim r \rightarrow p\) is

Consider a function $f: \mathbb{N} \to \mathbb{R}$, satisfying $f(1) + 2f(2) + 3f(3) + \cdots + xf(x) = x(x+1)f(x)$; $x \ge 2$ with $f(1)=1$. Then $\dfrac{1}{f(2022)} + \dfrac{1}{f(2028)}$ is equal to

JM Q34.

If the statements \((p \wedge \sim r) \rightarrow (q \vee r)\), \(q\) and \(r\) are all false, then \(p\)

Let a relation $R$ on $\mathbb{N}\times\mathbb{N}$ be defined as: $(x_1,y_1)\,R\,(x_2,y_2)$ if and only if $x_1\leq x_2$ or $y_1\leq y_2$. Consider the two statements: (I) $R$ is reflexive but not symmetric. (II) $R$ is transitive Then which one of the following is true?

Let X, Y be two sets and X has 40 elements, \(X \cup Y\) has 60 elements and \(X \cap Y\) has 10 elements. Then the number of elements in Y is

The real function \(f(x) = \cos^{-1}\sqrt{x^2 + 3x + 1} + \cos^{-1}\sqrt{x^2 + 3x}\) is defined on the set

If \(g(x)=x^2+x-1\) and \((g\circ f)(x)=4x^2-10x+5\), find \(f\!\left(\frac{5}{4}\right)\).

Let P(S) be the power set of S = {1, 2, . . . , 10}. Define A R1 B iff (A ∩Bc) ∪(B ∩Ac) = ∅, and A R2 B iff A ∪Bc = B ∪Ac. Then:

Let \(f:(-\infty,-2]\to[3,\infty)\), \(f(x)=x^2+2x+3\). Find \(f^{-1}(x)\).

Let $f^1(x) = \dfrac{3x+2}{2x+3}$, $x \in \mathbb{R} - \left\{-\dfrac{3}{2}\right\}$. For $n \ge 2$, define $f^n(x) = f^1 \circ f^{n-1}(x)$. If $f^5(x) = \dfrac{ax+b}{bx+a}$, $\gcd(a,b)=1$, then $a+b$ is equal to ______.

If \(f(\ln(1+|x|)) = (1 - \ln(1+|x|))^{\frac{1}{7}}\), then \(f(f(\cos x))\) is equal to:

JM Q23.

JM Q18 -- functional equation.

Let \(f(x) = \ln(x^2 + ax + 1)\). If \(f(x)\) is defined \(\forall\, x \in R\), then the number of integers in the range of \(a\) is:

Find the range of \(f(x) = \dfrac{\sec x + \tan x - 1}{\tan x - \sec x + 1}\), \(x \in \left(0, \dfrac{\pi}{2}\right)\)

Find the inverse of \(y=5^{\log x}\).

If \((a+1)(b+1)(c+1)(d+1) = 1\)\((a+2)(b+2)(c+2)(d+2) = 2\)\((a+3)(b+3)(c+3)(d+3) = 3\)\((a+4)(b+4)(c+4)(d+4) = 4\)Then find \((a+5)(b+5)(c+5)(d+5)\)

JM Q23.

JM Q24.

JM Q25.

JM Q27.

JM Q28.

JM Q29.

JM Q30.

JM Q31.

JM Q32.

JM Q34.