CAT Functions & Graphs questions (2017–2025) — actual Quant questions with solutions

A value of c for which the minimum value of f(x) = x² – 4cx + 8c is greater than the maximum value of g(x) = –x² + 3cx – 2c, is

Function f:N→W such that f(xy) = f(x)f(y) + f(x) + f(y) and f(p)=1 for primes. Find f(160000).

For non-zero real x, if f(x) + 2f(1/x) = 3x and f(x) = 3, find the sum of all such x.

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

The minimum possible value of x²−6x+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)=x². If f(x²−x)=5, then the value of the sum f(g(5))+g(f(5)) is

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 largest real value of a for which the equation |x+a|+|x−1|=2 has an infinite number of solutions for x is

For any real number x, let [x] be the largest integer less than or equal to x. If ∑Nn=1[15+n25]=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

If f(x)=x²−7x and g(x)=x+3 , then the minimum value of f(g(x))−3x is

For all real values of x, the range of the function f(x) = x²+2x+4/2x²+4x+9 is

f(x) = x²+2x−15/x²−7x−18 is negative if and only if

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

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

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

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

Consider a function f(x+y) = f(x) f(y) where x , y are positive integers, and f(1) = 2. If f (a+1) + f (a+2) + ..... + f(a+n) = 16 (2n − 1) then a is equal to.

Let f(x)=max{5x, 52 − 2x²}, where x is any positive real number. Then the minimum possible value of f(x) is

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

Let f(x) = x² and g(x) = 2x, for all real x. Then the value of f(f(g(x)) + g(f(x))) at x = 1 is

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

If f1(x) = x² + 11x + n and f2(x) = x, then the largest positive integer n for which the equation f1(x) = f2(x) has two distinct real roots, is:

If f(x) = 5x+2/3x−5 and g(x) = x² – 2x – 1, then the value of g(f(f(3))) is: