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 )

step-by-step explanation:

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is           symmetric about the line y = 120,000.

step-by-step explanation:

given that megan owns a clothing boutique.

the profits of 2010, 11, are recorded and shown as follows:

years since     2010 0     1     2     3       4       5

profits made in 1000's   39       84     111     120   111     84

we find that profit gradually increases form 2010 to 2013 upto 120,   then starts decreasing.

the profits for 2012 and 2014 are the same while that of 2011 and 2015 are also the same.

this shows that there is symmetry about the line profit = 120000

hence answer is:

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is           symmetric about the line y = 120,000.

read more on -

3 4 7 12 19

+1 +3 +5 +6

i would guess 26 because this isnt a sequential pattern or its to short to know