If `f` is an even function defined on the interval `(-5,5),` then four real values of `x` satisfying the equation `f(x)=f((x+1)/(x+2))` are ______, __________, _______ and______.

Options

A. 1
B. 2
C. 4
D. none of these

Answer

C

Explanation

Since f(x) is an even functions defined on `[-5,5]`.
`:. f(-x) =f(x)` for all ` x in [-5,5]`
Thus, `f(x)=f((x+1)/(x+2))`
`impliesx=(x+1)/(x+2) or , x=-(x+1)/(x+2)`
`implies x^(2)+x-1=0 or , x^(2)+3x+1=0`
`implies x=(-1 pmsqrt(5))/(2) or , s=(-3 pm sqrt(5))/(2)`
Hence, the value of x are
`(-1+sqrt(5))/(2),(-1-sqrt(5))/(2),(-3-sqrt(5))/(2) and (-3+sqrt(5))/(2)`
NOTE - Every real valued functions f(x) can be expressed uniquely as the sum of an even function `phi (x)=(1)/(2){f(x)+f(-x)}` and an odd function `Psi(x)=(1)/(2){f(x)-f(-x)}`.