Embibe Experts Solutions for Chapter: Functions, Exercise 3: EXERCISE-3

Let $f\left(x\right)=1-x-x^3$. If the set of values of $x$ satisfying the inequality, $1-f\left(x\right)-f^3\left(x\right)>f\left(1-5x\right)$ is $\left(a,0\right)\cup \left(b,\infty \right)$. Find $a+b$.

Suppose that $f$ is an even, periodic function with period $2$, and that $f\left(x\right)=x$ for all $x$ in interval $\left[0,1\right]$, then the value of $f\left(3.14\right)$, is

A function $f:\left[\frac{1}{2},\infty \right)\to \left[\frac{3}{4},\infty \right)$ defined as $f\left(x\right)=x^2-x+1$, then sum of all the solutions of the equation $f\left(x\right)=f^{-1}\left(x\right)$, is

Suppose $p\left(x\right)$ is a polynomial with integer coefficients. The remainder when $p\left(x\right)$ is divided by $x-1$ is $1$ and the remainder when $p\left(x\right)$ is divided by $x-4$ is $10$. If $r\left(x\right)$ is the remainder when $p\left(x\right)$ is divided by $\left(x-1\right)\left(x-4\right)$, then the value of $r\left(2006\right)$, is

Let $f\left(x\right)=\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)+5$ where $x\in \left[-6,6\right]$. If the range of the function is $\left[a,b\right]$ where $a, b\in N$, then the value of $\left(a+b\right)$, is

The set of real values of '$x$' satisfying the equality $\left[\frac{3}{x}\right]+\left[\frac{4}{x}\right]=5$ (where $\left[\right]$ denotes the greatest integer function) belongs to the interval $\left(a,\frac{b}{c}\right]$ where $a, b, c\in N$ and $\frac{b}{c}$is in its lowest form, then the value of $a+b+c+abc$ is

Let $f\left(x\right)=x^{135}+x^{125}-x^{115}+x^5+1$. If $f\left(x\right)$ is divided by $x^3-x$, then the remainder is some function of $x$ say $g\left(x\right)$, then the value of $g\left(10\right)$, is

Let $\left\{x\right\}$ & $\left\{x\right\}$ denote the fractional and integral part of a real number $x$, respectively, then number of solutions of the equations $4\left\{x\right\}=x+\left[x\right]$, is