Calculus question dealing with f o g

Question

Let ƒ(x)= {sqrt}-32 -x and g(x) = x^2 - 12x.

ƒ ∘ g =   

The domain of ƒ ∘ g is:

Answer 1

The composition is

(f ∘ g)(x)= √ (32 - x² + 12x)

The domain is the region where the expression below the square root is non-negative. To find this region, we first find where that expression is zero, that is, set:

32 - x² + 12x = 0

Using the quadratic formula, the roots are 6 + 2√17 and 6 - 2√17. So there are three regions to check:

-- Region 1: x < 6 - 2√17. Plugging in a very negative number, like x=-100, we see that the expression 32 - x² + 12x is negative.

-- Region 2: 6 - 2√17 < x < 6 + 2√17. Plugging x=0, we see that the expression 32 - x² + 12x is positive.

-- Region 3: x > 6 - 2√17. Plugging in a very large number, like x=100, we see that the expression 32 - x²+ 12x is negative.

Conclusion: the domain of the composition is the closed interval [6 - 2√17 , 6 + 2√17].