1. Find (f·g)(x) where f(x) = 1/x²+3 and g(x) = √x-2.
a) (f·g)(x) = 1/x-2
b) (f·g)(x) = 1/√x-2 +3
c) (f·g)(x) = √-2x²-5/x²+3
2. Find (g·f)(x) where f(x) = x²-2 and g(x) = 5x-8.
a) (g·f)(x) = 5x²-18
b) (g·f)(x) = 5x²-4
c) (g·f)(x) = 25x²+80x+62
d) (g·f)(x) = 5x²-10
3. Determine the domain of the function (f·g)(x) where f(x) = x²/x²-1 and g(x) = √x+4.
a) (-∞,-1)∪(-1,1)∪(1,∞)
b) (-4,-3)∪(-3,∞)
c) (-∞,-3)∪(-3,∞)
d) [-4,-3)∪(-3,∞)
and what are your answers?
To find (f·g)(x), we need to multiply the two functions f(x) and g(x) together and simplify the expression.
1. Find (f·g)(x) where f(x) = 1/x²+3 and g(x) = √x-2.
To multiply the two functions together, we substitute g(x) into f(x):
(f·g)(x) = f(x) * g(x) = (1/x²+3) * √x-2
To simplify this expression, we can use the properties of exponents and roots:
(f·g)(x) = (1/x²+3) * √x-2
= √(1/x²+3) * √(x-2)
= √((1/x²+3)*(x-2))
= √(x-2)/(x²+3)
So, the answer is (f·g)(x) = √(x-2)/(x²+3).
Therefore, the correct choice is option: c) (f·g)(x) = √(x-2)/(x²+3).
2. Find (g·f)(x) where f(x) = x²-2 and g(x) = 5x-8.
Similar to the previous question, we substitute f(x) into g(x):
(g·f)(x) = g(x) * f(x) = (5x-8) * (x²-2)
To simplify this expression, we multiply the terms using the distributive property:
(g·f)(x) = (5x-8) * (x²-2)
= 5x(x²-2) - 8(x²-2)
= 5x³ - 10x - 8x² + 16
The final expression is 5x³ - 8x² - 18x + 16.
Therefore, the correct choice is option: a) (g·f)(x) = 5x³ - 8x² - 18x + 16.
3. Determine the domain of the function (f·g)(x) where f(x) = x²/x²-1 and g(x) = √x+4.
The domain of a function is the set of all possible input values for which the function is defined. In this case, we need to find the values of x that make the denominator of f(x) and the radicand of g(x) valid.
For f(x) = x²/(x²-1), the denominator cannot be zero. Thus, we need to solve the equation x²-1 = 0:
x² - 1 = 0
(x - 1)(x + 1) = 0
This equation has solutions x = 1 and x = -1. So, the domain of f(x) is all real numbers except x = 1 and x = -1.
For g(x) = √x+4, the radicand (x + 4) must be non-negative, so x + 4 ≥ 0. Solving this inequality, we get x ≥ -4.
Therefore, the domain of (f·g)(x) is the intersection of the domains of f(x) and g(x), which is (-∞, -1) ∪ (-1, 1) ∪ (1, ∞).
Thus, the correct choice is option: a) (-∞, -1) ∪ (-1, 1) ∪ (1, ∞).