Determine the domain of (f/g)(x) when f(x) = 1/x and g(x) = sqrt(x+5).

I have the domains of both functions and I know how to set up (f/g)(x), but I don't know where to go from there.

(f/g)(x)= (1/x)/(sqrt(x+5))
domain of f(x) = (-∞,0)-(0,∞)
domain of g(x) = (-5,∞)

[1/x] / sqrt(x+5) yes =

= 1 /[ x sqrt(x+5) ]
that excludes x = 0 and x = -5 from the domain
-oo to -5
-5 to 0
0 to +oo

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Oh, so you're trying to perform some function division, huh? Like f divided by g, or as you wrote, (f/g)(x)? Well, let me help you out with that. Starting from what you've got, the domain of f(x) is all real numbers except for 0, and the domain of g(x) is all real numbers greater than or equal to -5.

Now, let's think about the domain of (f/g)(x). We know that we can't divide by 0, right? So we have to make sure that x ≠ 0. Additionally, the expression inside the square root in g(x), which is (x+5), must be greater than or equal to 0. So we have x+5 ≥ 0, which means x ≥ -5.

Putting it all together, the domain of (f/g)(x) is all real numbers except for 0, and also x must be greater than or equal to -5. So in fancy math language, we can say that the domain is (-∞, -5) U (-5, 0) U (0, ∞).

Hope that helps! Let me know if there's anything else I can clown around with for you! 🤡😄

To determine the domain of (f/g)(x), where f(x) = 1/x and g(x) = sqrt(x+5), we need to consider two things: the domains of both functions and any restrictions on the division operation.

The domain of f(x) = 1/x is given as (-∞,0) U (0,∞). This means that x can take any value except 0, as division by zero is undefined.

The domain of g(x) = sqrt(x+5) is given as (-5,∞). Here, x can take any value greater than or equal to -5, as the square root of a negative number is undefined.

Next, let's consider the division operation itself. For (f/g)(x) = (1/x)/(sqrt(x+5)), we need to ensure that the denominator is not equal to 0. In this case, the denominator is sqrt(x+5), so we need to make sure x+5 is not negative.

Considering all of this information, the domain of (f/g)(x) will be the intersection of the domains of f(x) and g(x), while also excluding any values of x that make the denominator equal to zero or negative.

Therefore, the domain of (f/g)(x) will be (-∞,-5) U (-5,0) U (0,∞). This means that x can take any value except those that make x+5 negative or x equal to zero.