g(x) = log3(x + 3)
state its domain, range, and asymptote.
i am having trouble with this can somebody help me?
remember we can take logs of only positive numbers, so ...
domain: x> -3
range : y is any real number
asymptote : x = -3
Of course, I can help you with that!
To find the domain of the function g(x) = log3(x + 3), we need to consider the restrictions on the logarithm function. The logarithm is defined only for positive real numbers, so the expression inside the logarithm (x + 3) must be greater than zero. Solving this inequality, we have:
x + 3 > 0
Subtracting 3 from both sides, we get:
x > -3
Therefore, the domain of g(x) is all real numbers greater than -3, or (-3, ∞).
Moving on to the range, we consider the logarithm function's behavior. The range of the logarithm function is all real numbers. Hence, the range of g(x) is (-∞, ∞).
Finally, let's determine the asymptote of the function. Since g(x) is a logarithmic function with a base of 3, the vertical asymptote occurs when the argument of the logarithm (x + 3) approaches zero. Therefore, x + 3 = 0, which implies that the vertical asymptote is x = -3.
So, to summarize:
- Domain: (-3, ∞)
- Range: (-∞, ∞)
- Asymptote: x = -3