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,∞)
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
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.
To solve these questions, we need to understand how to perform function composition.
1. To find (f*g)(x) where f(x) = 1/(x^2+3) and g(x) = sqrt(x-2), we need to substitute g(x) into f(x).
Start by replacing x in f(x) with g(x):
f(g(x)) = 1/((g(x))^2 + 3)
f(g(x)) = 1/((sqrt(x-2))^2 + 3)
Simplify the expression inside the parentheses:
f(g(x)) = 1/((x-2) + 3)
f(g(x)) = 1/(x + 1)
So the correct answer is (c) (f*g)(x) = 1/(x + 1).
2. To find (g*f)(x) where f(x) = x^2-2 and g(x) = 5x-8, we substitute f(x) into g(x):
g(f(x)) = 5(f(x)) - 8
g(f(x)) = 5(x^2 - 2) - 8
Distribute and simplify:
g(f(x)) = 5x^2 - 10 - 8
g(f(x)) = 5x^2 - 18
So the correct answer is (a) (g*f)(x) = 5x^2 - 18.
3. To determine the domain of the function (f*g)(x) where f(x) = x^2/(x^2-1) and g(x) = sqrt(x+4):
The domain of a function is the set of all the possible values of x for which the function is defined.
Since we have a radical function g(x) = sqrt(x+4), the argument (x+4) inside the square root cannot be negative.
Solve the inequality:
x + 4 >= 0
x >= -4
So the domain for g(x) is x >= -4.
For f(x) = x^2/(x^2-1), we need to exclude the values of x that make the denominator zero, since division by zero is undefined.
Solve the equation x^2 - 1 = 0:
(x+1)(x-1) = 0
x = -1 or x = 1
Therefore, the domain for f(x) is x ≠ -1, 1.
To determine the domain of (f*g)(x), we need to find the intersection of the domains of f(x) and g(x), which is the range of x values that are common to both functions.
The domain of (f*g)(x) is the intersection of x >= -4 (from g(x)) and x ≠ -1, 1 (from f(x)).
After considering the ranges for x, option (b) (-4, -3)∪(-3,∞) is the correct answer.
By breaking down the composition of functions and considering the domain restrictions, we can find the correct answers to these questions.