Functions CAT PYQs , 40 Questions with Solutions

Let $f(x) = \dfrac{x}{(2x-1)}$ and $g(x) = \dfrac{x}{(x-1)}$. Then, the domain of the function $h(x) = f(g(x)) + g(f(x))$ is all real numbers except

For real values of $x$, the range of the function $f(x) = \frac{2x-3}{2x^2+4x-6}$ is

Let $x, y$, and $z$ be real numbers satisfying $\begin{aligned} & 4\left(x^2+y^2+z^2\right)=a \\ & 4(x-y-z)=3+a \end{aligned}$ Then $a$ equals

Consider two sets $A=\{2,3,5,7,11,13\}$ and $B=\{1,8,27\}$ . Let $f$ be a function from $A$ to $B$ such that for every element $b$ in $B$ , there is at least one element $a$ in $A$ such that $f(a)=b$ . Then, the total number of such functions $f$ is

A function $f$ maps the set of natural numbers to whole numbers, such that $f(x y)=f(x) f(y)+f(x)+f(y)$ for all $x, y$ and $f(p)=1$ for every prime number $p$ . Then, the value of $f(160000)$ is

For any non-zero real number $x$ , let $f(x)+2 f\left(\frac{1}{x}\right)=3 x$ . Then, the sum of all possible values of $x$ for which $f(x)=3$ , is

Suppose $f(x, y)$ is a real-valued function such that $f(3 x+2 y, 2 x-5 y)=19 x$ , for all real numbers $x$ and $y$ . The value of $x$ for which $f(x, 2 x)=27$ , is

Let $0 \leq a \leq x \leq 100$ and $f(x)=|x-a|+|x-100|+|x-a-50|$ . Then the maximum value of $f(x)$ becomes 100 when $a$ is equal to

Consider the real-valued function $f(x) = \frac{\log(3x - 7)}{\sqrt{2x^2 - 7x + 6}}.$ Find the domain of $f(x).$

Suppose for all integers $x$, there are two functions $f$ and $g$ such that $f(x) + f(x-1) - 1 = 0$ and $g(x) = x^2$. If $f\left(x^2 - x\right) = 5$, then the value of the sum $f(g(5)) + g(f(5))$ is

Find the minimum value of $,\dfrac{x^2 - 6x + 10}{3 - x},$ for $x < 3$

Let $r$ be a real number and $f(x)=\left\{\begin{array}{cl}2 x-r & \text { if } x \geq r \\ r & \text { if } x\lt r\end{array}\right.$ . Then, the equation $f(x)=f(f(x))$ holds for all real values of $x$ where

f(x) = $\frac{x^{2}+2 x-15}{x^{2}-7 x-18}$ is negative if and only if

Which of the following graphs represent the function of x?

Graph A: a smooth horizontal wavy curve crossing the x-axis a few times near the origin (like a portion of a sine wave lying on its side, extending along the x-axis).

Graph B: a vertical S-shaped squiggle lying entirely to the right of the y-axis, oriented vertically (it curves back on itself along the y-direction rather than the x-direction).

Graph C: a circle of radius 3 centered at the origin, labeled f(x) on the y-axis.

Graph D: a V-shape made of two rays starting from the origin, one going up-left and one going up-right (like y = |x|).

Choose the correct answer from the options given below:

If x₀ = 1, x₁ = 2, and $x_{n+2} = \frac{1 + x_{n+1}}{x_n}$, $n = 0, 1, 2, 3, \ldots$, then $x_{2021}$ is equal to?

Which of the following equations best describes the graph given below? (The graph shows two mirrored V-shaped wedge curves: one opening rightward with vertex at (6,0) and branches passing through (8,2) and (8,-2); a mirrored one opening leftward with vertex at (-6,0) and branches passing through (-8,2) and (-8,-2).)

For all real values of x, the range of the function f(x) = $\frac{x^{2} + 2x + 4}{2x^{2} + 4x + 9}$ is

If $f(x)=x^{2}-7 x$ and $g(x)=x+3$ , then the minimum value of $f(g(x))-3 x$ is

Rahul has just made a 3 × 3 magic square, in which, the sum of the cells along any row, column or diagonal, is the same number N. The entries in the cells are given as expressions in x, y, and z. Find N?

If f(5 + x) = f(5 - x) for every real x and f(x) = 0 has four distinct real roots, then the sum of the roots is

The number of real-valued solutions of the equation $2^x + 2^{-x} = 2 - (x - 2)^2$ is

For real $x$, the maximum possible value of $\frac{x}{\sqrt{1 + x^{4}}}$ is

Let $k$ be a constant. The equations $kx + y = 3$ and $4x + ky = 4$ have a unique solution if and only if

If x₁ = -1 and xₘ = xₘ₊₁ + (m + 1) for every positive integer m, then x₁₀₀ equals

If f(x+y) = f(x)f(y) and f(5) = 4, then f(10) - f(-10) is equal to

If A ⊙ B = $(A + B) \times B$, then what is $(5 ⊙ 2) ⊙ 5$?

The number of the real roots of the equation 2cos(x(x + 1)) = 2 x + 2 -x is

For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8f(m + 1) - f(m) = 2, then m equals [TITA]

Consider a function $f(x+y) = f(x) \times f(y)$ where $x$, $y$ are positive integers, and $f(1) = 2$. If $f(a+1) + f(a+2) + \text{...} + f(a+n) = 16 (2^n - 1)$ then $a$ is equal to.

Let f be a function such that f(mn) = f(m) f(n) for every positive integers m and n. If f(1), f(2) and f(3) are positive integers, f(1) < f(2), and f(24) = 54, then f(18) equals [TITA]

Let f(x)=min{2x2, 52 - 5x}, where x is any positive real number. Then the maximum possible value of f(x) is [TITA]

If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = 91, f(15) = 617, then f(10) equals. [TITA]

Let $f(x) = \min\{2x^2,\ 52 - 5x\}$ where $x$ is a positive real number. Then maximum possible value of $f(x)$?

Let $f(x) = \max\{5x, 52 - 2x^2\}$, where $x$ is any positive real number. Then the minimum possible value of $f(x)$ is (TITA).

If f(x) = (x+1)/(x-1), the ratio of x to f(y), where y = f(x) is

Output is produced according to Q=(a-4L)^6, where 'L' is the quantity of labour input and 'a' is the quantity of capital input. Then, which of the following alternative is correct?

If $f(x) = \dfrac{5x + 2}{3x - 5}$ and $g(x) = x^2 - 2x - 1$, then the value of $g(f(f(3)))$ is:

Let f(x) = 2x – 5 and g(x) = 7 – 2x. Then |f(x) + g(x)| = |f(x)| + |g(x)| if and only if

If f(ab) = f(a)f(b) for all positive integers a and b, then the largest possible value of f(1) is [TITA]

Let $f(x) = x^2$ and $g(x) = 2x$, for all real $x$. Then the value of $f(f(g(x)) + g(f(x)))$ at $x = 1$ is