Function NDA Pyqs

00:00 Introduction 0:45 - If $2f(x) + f(1 - x) = x$, then what is $f(x)$ equal to? 04:18 - Given a quadratic polynomial $f(x) = ax^2 + bx + c$ such that $f(1) = f(4) = 2$ and $2$ is a root of $f(x) = 0$. 05:05 - What is the value of $a + b + c$? 07:42 - What is the other root of $f(x) = 0$? 09:27 - Given $f(x) = \frac{x}{1-x}$, where $x$ is positive and $x \neq 1$. 10:30 - What is the value of $\frac{f(x)}{f(x+1)}$? 11:22 - What is the value of $(1 - x)f(x) + xf(x + 1)$? 12:27 - What is the domain of the function $f(x) = 1 - (x - 1)^{2/3}$? 14:18 - Given $f(x) = \frac{10^x - 10^{-x}}{10^x + 10^{-x}}$. 14:40 - What is the value of $(f \circ f \circ f \circ f \circ f)(0)$? 15:43 - What is the inverse of the function? 18:19 - What is the range of the function $y = (1 - \cos x)^{-1}$, where $x \neq 2n\pi$? 20:27 - What is the domain of the function $f(x) = |x - 3|$? 21:58 - Given $f = {(1, 1), (2, 4), (3, 7), (4, 10)}$. 22:32 - If $f(x) = px + q$, what is the value of $p + q$? 23:05 - Is the function one-to-one and onto given the co-domain is the set of natural numbers? 25:35 - A function $f(x)$ satisfies $f(x/y) = f(x)/f(y)$ for all positive real values of $x$ and $y$, and $f(2) = 3$. 26:58 - What is the value of $f(1) \cdot f(4)$? 27:10 - What is the value of $f(16)$? 28:09 - A function $f$ is such that $f(xy) = f(x) + f(y)$ and $f(5) = 10$. 28:30 - What is $f(0)$? 28:53 - What is $f(20) + f(-20)$? 30:09 - If $f(x) = 4x + 3$, then what is the value of $f(f(f(-1)))$? 30:45 - If $4f(x) - f(1/x) = (2x + 1/x)(2x - 1/x)$, what is the value of $f(2)$? 32:30 - What is the domain of the function $f(x) = \sqrt{1 - (x - 1)^2}$? 33:57 - Which of the following is correct for $f(x) = \ln(x + \sqrt{1 + x^2})$? (Discussion on even and odd functions). 38:32 - If $f(x) = 4x + 1$, $g(x) = kx + 2$, and $f \circ g = g \circ f$, what is the value of $k$? 39:26 - Let $z = [y]$ and $y = [x] - x$, where $[ \cdot ]$ is the greatest integer function. If $x$ is not an integer but positive, what is the value of $z$? 42:33 - Given $f(x + y) = f(x)f(y)$ and $f(1) = 2$. 44:16 - Find the value of $n$ if the sum from $x=2$ to $n$ of $f(x) = 2044$. 46:51 - Given $A = {7, 8, 9, \dots, 16}$ and $f: A \to \mathbb{N}$ is defined by $f(x) = $ highest prime factor of $x$. How many elements are in the range? 48:01 - Discussion on whether specific relations $f = {(xy, x - y) \mid x, y \in \mathbb{Z}}$ and $f = {(xy, x + y) \mid x, y \in \mathbb{N}}$ are functions between the given sets. 51:09 - If $f(x) = x(4x^2 - 3)$, what is $f(\sin \theta)$? 51:45 - If $f(x) = x^2 + 2$ and $g(x) = 2x - 3$, what is $fg(1)$? 52:30 - What is the range of $f(x) = x + |x|$? 55:12 - Given $f(x) = \sin([\pi^2]x) + \cos([-\pi^2]x)$, what are the values of $f(\pi/2)$ and $f(\pi/4)$? 59:32 - If $f: A \to B$ is defined as $f(x) = \frac{2x + 3}{3x + 5}$, and $f$ is onto, what are $A$ and $B$? 01:03:24 - What is the domain of $f(x) = \sqrt{2 - x} + \sqrt{2 + x}$? 01:04:21 - What is the range and period of $f(x) = \pi + \sin^2 x$?