PLEASE HELP If g(x)=1+x(x∈ ℝ ) and h(x)=x2+2x(x∈ ℝ ), find the ranges of g and h. Find the composite functions g∘h and h∘g, stating their ranges.

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 )