let f(x)=sqrt(x-3), g(x)=sqrt(x^2+3)
Find a formula for f*g and state the domain of the composition
f(x) = sqrt(x-3)
g(x) = sqrt(x²+3)
f*g(x) (composition)
= f(g(x))
= f(sqrt(x²+3))
= sqrt(sqrt(x²+3)-3)
The domain of g(x) is (-∞,∞)
and the domain of f(x) is [3,∞).
Therefore the domain of f*g(x) is such that
g(x)≥3.
sqrt(x²+3) ≥ 3
x²+3 ≥ 9
x² ≥ 6
x≥√6 or x≤-√6
Therefore the domain of f*g(x) is
(-∞,-√6]∪[√6,∞)
To find the formula for f*g, we need to perform the composition of the two functions f(x) and g(x).
The composition of two functions f(x) and g(x) is denoted as (f ∘ g)(x) and is defined as f(g(x)). Therefore, (f ∘ g)(x) = f(g(x)).
Given f(x) = √(x - 3) and g(x) = √(x^2 + 3), we can substitute g(x) into f(x) to find (f ∘ g)(x).
(f ∘ g)(x) = f(g(x)) = f(√(x^2 + 3))
Now, let's substitute g(x) = √(x^2 + 3) into f(x):
f(g(x)) = √(g(x) - 3)
Substituting g(x) = √(x^2 + 3) into the above expression:
(f ∘ g)(x) = √(√(x^2 + 3) - 3)
So, the formula for (f ∘ g)(x) is √(√(x^2 + 3) - 3).
Now, let's determine the domain of the composition (f ∘ g)(x).
To find the domain, we need to consider the restrictions on the square root function and the composition of the two functions.
For the square root function, the radicand (expression inside the square root) must be greater than or equal to zero. So, for √(x^2 + 3) to be defined, the expression x^2 + 3 must be greater than or equal to zero.
x^2 + 3 ≥ 0
x^2 ≥ -3
The square of any real number is always greater than or equal to zero. Therefore, x^2 is always greater than or equal to zero.
Since x^2 ≥ 0 for all real numbers, the inequality x^2 ≥ -3 holds true for all values of x.
Hence, the domain of (f ∘ g)(x) is all real numbers (-∞, ∞), as there are no restrictions on x for the composition of these two functions.
To find a formula for f*g, we need to find the composition of f and g, denoted as "f(g(x))".
Substituting g(x) into f(x), we have:
f(g(x)) = f(sqrt(x^2 + 3))
Now, let's simplify further by substituting f(x) into this expression:
f(g(x)) = sqrt(g(x) - 3)
= sqrt(sqrt(x^2 + 3) - 3)
So, the formula for f*g is f(g(x)) = sqrt(sqrt(x^2 + 3) - 3).
To determine the domain of the composition, we need to consider the individual domains of f(x) and g(x) and find the intersection of those domains.
For f(x) = sqrt(x - 3), x - 3 must be greater than or equal to zero (since we cannot take the square root of a negative number).
So, the domain of f(x) is x ≥ 3.
For g(x) = sqrt(x^2 + 3), the expression inside the square root, x^2 + 3, can be any real number since any real number squared is always non-negative.
Therefore, the domain of g(x) is -∞ < x < ∞.
To determine the domain of the composition f(g(x)), we need to find the intersection of the domains of f(x) and g(x).
Since the domain of g(x) is all real numbers, and the domain of f(x) is x ≥ 3, the domain of their composition f(g(x)) will be the intersection of these two domains.
Hence, the domain of f(g(x)) is x ≥ 3.
Therefore, the formula for f*g is f(g(x)) = sqrt(sqrt(x^2 + 3) - 3), and the domain of the composition is x ≥ 3.