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Question
Show that the function f: ℝ → ℝ defined by f(x) = `x/(x^2 + 1), ∀x in R`is neither one-one nor onto. Also, if g: ℝ → ℝ is defined as g(x) = 2x - 1. Find fog(x)
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Solution
Given `y = x/(x^2+1)`
`=> yx^2 - x + y = 0`
Here a = y, b = -1 and c = y
`:. x = (-(-1)+- sqrt(1-4y^2))/(2y)`
Clearly for every value of y, x will have two different values so the function is many−one not one−one
Since `1 -4y^2 >= 0 => (1+2y)(1-2y)>= 0 => (-1)/2 <= y ><= 1/2`
That means no matter what is x, y always belongs to the interval `[(-1)/2, 1/2]`
So, the function is not onto
Now, fog(x) = `(2x-1)/((2x-1)^2 +1) = (2x+1)/(4x^2 - 4x + 1+1) = (2x+1)/(2(2x^2 - 2x + 1))`
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