Given the function below, find the domain of (f∘g)(x)

Question

f(x)=1/13x−26

g(x)=root2x+24

How to find the domain

Answer 1

I believe f(x) = 1/(13x - 26) = 1/(13(x-2))

Then, f(x) is defined for all x ≠ 2

g(x) = √(2x+24) is defined for 2x+24 >= 0, or x >= -12

Then, f(g(x)) is defined for g(x) ≠ 2

√(2x+24) = 2 when 2x+24 = 2² , or 2x + 24 = 4, or 2x = -20, or x = -10

Thus, the domain is [-12, 10) ∪ (10, ∞)

Answer 2

With composite functions you need to look at both the "inner" function and the composite function.

f(x) = (1/13)x -26

g(x) = sqrt (2x+24)

(f°g)(x) = (1/13)sqrt(2x+24) - 26

There are 2 main areas we need to look at when determining the domain.

1) the denominator, if one exists, cannot equal zero, since you cannot divide by zero.

2) the radicand, the expression inside the square root, cannot be negative. You cannot take the square root of a negative number. This applies not only to square roots but any even root.

The first item does not apply in this problem since there is no denominator. However, the second item does.

so 2x+24 ≥ 0

solving for x

2x ≥ -24

x ≥ -12

In interval notation, [-12,∞)