Find (f*g)(x) and its domain when f(x)=x^2+9 and g(x)= sqrt x+3
A. (f·g)(x)=x+12; (-∞,∞)
B. (f·g)(x)=√x^2+12; (-∞,∞)
C. (f·g)(x)=√x^2+12; [-3,∞)
D. (f·g)(x)=x+12; [-3,∞)
I need help please. I don't know how to get the correct answer
>sigh<
No. What happens if you try to take f(g(-10))?
g(-10) is undefined, so there's no way to evaluate f.
So, D is the right choice. The domain of g helps to determine the domain of f(g)
Well, let's break it down step by step!
First, we need to find f * g, which means we need to multiply the two functions together.
f(x) = x^2 + 9
g(x) = √(x+3)
To find (f * g)(x), we substitute g(x) into f(x):
(f * g)(x) = f(g(x))
= f(√(x+3))
= (√(x+3))^2 + 9
= (x+3) + 9
= x + 12
So, (f * g)(x) = x + 12.
Next, let's find the domain of (f * g)(x).
The domain of (f * g)(x) is the set of x-values for which the function is defined.
Since both f(x) = x^2 + 9 and g(x) = √(x+3) are defined for all real numbers, the domain of (f * g)(x) = (-∞,∞).
So, the correct answer is option A. (f · g)(x) = x + 12; (-∞,∞).
To find the product (f*g)(x) and its domain, we need to multiply the two functions f(x) and g(x) and determine the resulting function and its domain.
1. Start with the given functions:
f(x) = x^2 + 9
g(x) = √(x + 3)
2. Multiply the two functions together:
(f*g)(x) = f(x) * g(x) = (x^2 + 9) * √(x + 3)
3. Simplify the expression by distributing the multiplication:
(f*g)(x) = x^2 * √(x + 3) + 9 * √(x + 3)
4. Now, let's determine the domain of the resulting function. Since we have a square root in the expression, we need to ensure that the radicand (x + 3) is non-negative (≥ 0).
For the square root to be defined, we have:
x + 3 ≥ 0
This inequality tells us that x must be greater than or equal to -3.
So, the domain of the function is [-3, ∞).
Now, let's examine the answer choices:
A. (f·g)(x) = x + 12; (-∞, ∞)
B. (f·g)(x) = √(x^2 + 12); (-∞, ∞)
C. (f·g)(x) = √(x^2 + 12); [-3, ∞)
D. (f·g)(x) = x + 12; [-3, ∞)
Comparing the simplified expression we obtained [(f·g)(x) = x^2 * √(x + 3) + 9 * √(x + 3)], none of the answer choices match.
Therefore, none of the given answer choices are correct.
f◦g = f(g) = g^2 + 9 = √(x+3)^2 + 9 = x+3 + 9 = x+12
So, it's gonna be either A or D.
But g(x) is undefined for x < 3, so ...