Answer:
y = g(x) = 1 + x
Now, the range of a function is the set of the possible values of y.
In this function, a linear function, y can be any real number, so the range of this function is { y ∈ ℝ )
y = h(x) = x^2 + 2*x
This is a quadratic function, as the leading coefficient is positive, we know that the arms of the function go up.
For a quadratic function y = a*x^2 + b*x + c
The minimum is at:
x = -b/2a.
In this case, b = 2 and 1 = 1
the minimum is at:
x = -2/2 = -1
The minimum is:
y = h(-1) = 1^2 +2*(-1) = -1
Then the range of this function is
{y ∈ ℝ ≥ -1 )
The composite functions are:
g∘h = g(h(x)) = 1 + x^2 + 2*x
The minimum is still at x = -1
g∘h(-1) = 1 + -1^2 + 2*-1 = 0
Then the range of this function is:
{y ∈ ℝ ≥ 0 )
The other composition is:
h∘g = h(g(x)) = (1 + x)^2 + 2*(1 + x) = 1 + 2*x + x^2 + 2 + 2*x
h∘g = x^2 + 4*x + 3
Here the minimum is at:
x = -4/2*1 = -2
h∘g(-2) = (-2)^2 + 4*-2 + 3 = 4 - 8 + 3 = -1
The range is:
{y ∈ ℝ ≥ -1 )
