Question

I really need help on this as I have tried multiple times and my answers are none of these. Please help

1. Find (f*g)(x) where f(x)=1/(x^2+3) and g(x)=sqrt(x-2).
a. (f*g)(x)=1/x-2
b. (f*g)(x)=1/sqrt(x-2)+3
c. (f*g)(x)=1/x+1
d. (f*g)(x)=sqrt(-2x^2-5/x^2+3)

2. Find (g*f)(x) where f(x)=x^2-2 and g(x)=5x-8.
a. (g*f)(x)=5x^2-18
b. (g*f)(x)=5x^2-4
c. (g*f)(x)=25x^2+80x+62
d. (g*f)(x)=5x^2-10

3. Determine the domain of the function (f*g)(x) where f(x)=x^2/x^2-1 and g(x)=sqrt(x+4).
a. (-∞,-1)∪(-1,1)∪(1,∞)
b. (-4,-3)∪(-3,∞)
c. (-∞,-3)∪(-3,∞)
d. [-4,-3)∪(-3,∞)

Answer 1

for functions a(x) and b(x)

(a*b)(x) = a[b(x)]

1. (f*g)(x) = 1 / {[g(x)]^2 + 3}
... = 1 / {[√(x-2)]^2 - 3}
... = 1 / [(x - 2) + 3] = 1 / (x + 1)

substitute the 2nd function in place of x in the 1st function

Answer 2

For #3,
(f◦g)(x) = f(g) = g^2/(g^2-1) = (x+4)/(x+3)

Since f(g) is not defined for g^2=1
x = -3 must be excluded
g(x) is undefined when x < -4
x = 1,-1 are in the domain of f◦g even though they are not in the domain of f.