ALGEBRA HELPP!!!.

Question

I am confused about this question because of ln, can someone explain these in steps?

G(x) = ln (1−x^2) and f(x) =√(x+3), what is the subset of the domain of g at the composition of f(g(x))? Domain of g◦f? What are (f◦g) (x) and (g◦f) (x)?

I will really appreciate it...

Answer 1

g(x)= ln(1 − x²) and f(x) =√(x+3)

Domain of f◦g

(f◦g)(x) means f(g(x)) = √(ln(1 − x²) + 3)

The presence of ln(1 − x²) in the function limits the domain to –1 < x < 1

because of the requirement 1 − x² > 0 (you can only take the logarithm of a positive number)

In interval notation (in case it's required), the domain of f◦g is (–1,1)

Domain of g◦f

(g◦f)(x) means g(f(x)) = ln(1 − [f(x)]²) and [f(x)]²= [√(x+3)]² = x + 3

So (g◦f)(x) = ln(1 − (x + 3)) = ln(–x – 2)

The requirement that –x – 2 > 0 means the x < –2

In interval notation (in case it's required), the domain of g◦f is (–∞,–2)