give the domain of the function and identify any vertical or horizontal asymptotes.
g(x)= 1 / x-3 + 1
My answer was x = o and y=3 but this was wrong. Please explain!!
If the question is
g(x) = 1/(x-3) + 1 then
the vertical asymptote is x = 3
and the horizontal is y = 1
(I believe Damon asked you yesterday for a clarification about the confusing denominator, but today you typed it exactly the same way.)
There are no parentheses used in the equation. Its just 1 over x-3 and then plus 1 is on the right side.
when you say in "words"
"Its just 1 over x-3 and then plus 1" we know exactly what it means, namely
1/(x-3) + 1 as I guessed
but when you write
1/x - 3 + 1, the 1 is only divided by the x and then 3 is subtracted.
In your text it probably has a fraction bar to show the fraction. Since we cannot create a "fraction bar" in this format, we have to use brackets to clarify what we mean.
To identify the domain of the function and any vertical or horizontal asymptotes, let's analyze the function g(x) = 1/(x-3) + 1.
1. Domain:
The domain of a function is the set of all real numbers for which the function is defined. In this case, the function g(x) is defined for all values of x except where the denominator (x-3) is zero. So, to find the domain, we need to set the denominator equal to zero and solve for x:
x - 3 = 0
x = 3
Therefore, the function is undefined when x = 3. Hence, the domain of g(x) is all real numbers except x = 3.
Domain: (-∞, 3) U (3, +∞)
2. Vertical Asymptote:
A vertical asymptote occurs when the function approaches positive or negative infinity as x approaches a certain value. In this case, there is a vertical asymptote when the denominator (x - 3) approaches zero. As we found earlier, the function is undefined at x = 3. Therefore, there is a vertical asymptote at x = 3.
Vertical asymptote: x = 3
3. Horizontal Asymptote:
A horizontal asymptote represents the value that the function approaches as x approaches positive or negative infinity. To determine the horizontal asymptote, we examine the behavior of the function as x becomes increasingly large or small.
In our case, as x approaches positive or negative infinity, the term 1/(x-3) becomes negligible compared to the constant 1. Therefore, the horizontal asymptote is y = 1.
Horizontal asymptote: y = 1
In summary,
Domain: (-∞, 3) U (3, +∞)
Vertical asymptote: x = 3
Horizontal asymptote: y = 1