Given the function f(x)=2(x+1)/(x^2+2x+5) determine the image of the function.

The image of the function is the set of elements f(x),
where the values of x belong to the domain of the
function.

Let y =
2(x+1)/(x^2+2x+5)

y*(x^2+2x+5) = 2x +
2

y*x^2 + 2y*x + 5y - 2x - 2 =
0

We'll consider y as a coefficients and we'll combine like
terms in x:

y*x^2 + x*(2y - 2) + 5y - 2 = 0
(1)

Since the values of x are considered as being real
numbers, we'll impose the constraint that the discriminant of the equation (1) to be
positive or, at least, zero.

delta = (2y-2)^2 -
4y*(5y-2)>=0

delta = 4y^2 - 8y + 4 - 20y^2 +
8y

We'll combine and eliminate like
terms:

delta = -16y^2 + 4 >=
0

We'll divide by -16:

y^2 -
4/16 =< 0

We'll recognize the difference of 2
squares:

(y - 2/4)(y+2/4) =<
0

(y - 1/2)(y + 1/2) =<
0

Since the expression y^2 - 4/16 is negative
over the range [-1/2 ; 1/2], then the image of the function f(x) is the set [-1/2 ;
1/2].