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Let f and g be real - valued functions s...

Let f and g be real - valued functions such that `f(x+y)+f(x-y)=2f(x)*g(y), forall " x , y " in R.` Prove that , if f(x) is not identically zero and `abs(f(x)) le 1, forall x in R,` then `abs(g(y)) le 1, forall y in R.`

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ARIHANT MATHS-FUNCTIONS-Exercise (Questions Asked In Previous 13 Years Exam)
  1. Let f and g be real - valued functions such that f(x+y)+f(x-y)=2f(x)*g...

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  2. If function f(x)=x^(2)+e^(x//2) " and " g(x)=f^(-1)(x), then the value...

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  3. Let F(x) be an indefinite integral of sin^(2)x Statement-1: The fun...

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  4. Find the range of values of t for which 2sint=(1-2x+5x^2)/(3x^2-2x-1)

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  5. Let fk(x) = 1/k(sin^k x + cos^k x) where x in RR and k gt= 1. Then f4(...

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  6. The function f:[0,3] to [1,29], defined by f(x)=2x^(3)-15x^(2)+36x+1 i...

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  7. Let f(x)=x^2a n dg(x)=sinxfora l lx in Rdot Then the set of all x sat...

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  8. Let f:(0,1)->R be defined by f(x)=(b-x)/(1-bx), where b is constant s...

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  9. Let f be a real-valued function defined on the inverval (-1,1) such th...

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  10. If X and Y are two non-empty sets where f: X->Y,is function is define...

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  11. If f(x)={x, when x is rational and 0, when x is irrational g(x)={0, wh...

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  12. If f(x)=sinx+cosx, g(x)=x^(2)-1, then g{f(x)} is invertible in the dom...

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  13. Domain of definition of the function f(x)=sqrt(sin^(-1)(2x)+pi/6) fo...

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  14. The range of the function f(x)=(x^2+x+2)/(x^2+x+1),x in R , is (1,oo)...

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  15. If f:[0,infty) rarr [0,infty) " and " f(x)=x/(1+x), then f is

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  16. If f:R to R be defined by f(x) =2x+sinx for x in R, then check the na...

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  17. Let E={1,2,3,4}a n dF-{1,2}dot If N is the number of onto functions fr...

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  18. Suppose f(x)=(x+1)^2forxgeq-1. If g(x) is the function whose graph is ...

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  19. If f:[1,infty) rarr [2,infty) is given by f(x)=x+1/x, " then " f^(-1)(...

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  20. Let f(x0=(1+b^(2))x^(2)+2bx+1 and let m(b) be the minimum value of f(x...

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  21. The domain of definition of function of f(x)=(log(2)(x+3))/(x^(2)+3x+2...

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