If `f(x)=(|x|)/(x)` and c be a non-zero real number, show that `|f(c)-f(-c)|=2.`

If f(x)=e^(px+q) [p and q are constants], show that, f(a).f(b).f(c)=f(a+b+c).e^(2q)

If f(x+y)=f(x)+f(y) for all real x and y, show that f(x)=xf(1) .

If f:R rarr R be defined by f(x)=x^2+3 where R is the set of real numbers then show that f^(-1)(7)={-2,2} and f^(-1)(19)={-4,4}

If 2f(x) + 3f(-x) = 15 - 4x for all real values of x , then show that f(x) = 3 + 4x.

Show that the function f defined by f(x)= |1-x+|x|| , where x is any real number, is a continuous function.

Let f(x)=x+2|x+1|+2|x-1|dot If f(x)=k has exactly one real solution, then the value of k is (a) 3 (b) 0 (c) 1 (d) 2

If f(x)=cos^2x+s e c^2x ,then

let h(x)=f(x)-[f(x)]^(2)+[f(x)]^(3) for every real number x. Prove that h(x) is increasing or decreasing according as f(x) is increasing or decreasing.

If f(x) is a twice differentiable function such that f(a)=0, f(b)=2, f(c)=-1,f(d)=2, f(e)=0 where a < b < c < de, then the minimum number of zeroes of g(x) = f'(x)^2+f''(x)f(x) in the interval [a, e] is

Let f(x) = Ax^2+Bx+C where A,B,C are real numbers. Prove that if f(x) is an integer whenever x is an integer, then the number 2A, A+B and C are all integers. Conversely, prove that if the numbers 2A, A+B and C are all integer then f(x) is an integer whenever x is an integer.