List of practice Questions

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

A function f maps the set of natural numbers to whole numbers, such that f(xy) = 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

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

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

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

Let \(f(x)\) be a quadratic polynomial in x such that \(f(x)≥0\) for all real numbers x.If \(f(2)=0\) and \(f(4)=6\),then \(f(−2)\) is equal to

The minimum possible value of \(\frac{x^2−6x+10}{3−x}\),for x<3,is

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 \(f(x^2−x)=5\),then the value of the sum \(f(g(5))+g(f(5))\) is

For any real number x, let [x] be the largest integer less than or equal to x. If \(\sum_{n=1}^N \left[ \frac{1}{5} + \frac{n}{25} \right] = 25\) then N is

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