Give the domain of the function and identify any vertical or horizontal asymptotes.
h(x) = 1 / (x - 3) + 1
I had x = 0 and y = 3. Would the domain be all numbers except 1?
To find the domain of the function h(x) = 1 / (x - 3) + 1, we need to consider any values of x that would make the denominator (x - 3) equal to zero since division by zero is undefined.
Setting the denominator equal to zero, we have:
x - 3 = 0
Solving for x, we get:
x = 3
So, x = 3 is not in the domain, and you are correct that all other numbers are part of the domain. Therefore, the domain of the function h(x) is all real numbers except x = 3.
Now let's identify any vertical and horizontal asymptotes for the function.
A vertical asymptote occurs when the function approaches infinity or negative infinity as x approaches a certain value. In this case, since the denominator is (x - 3), the function will have a vertical asymptote at x = 3. As x approaches 3 from both sides (x < 3 and x > 3), the function will become not defined (division by zero).
To determine if there is a horizontal asymptote, we can investigate the behavior of the function as x approaches positive infinity and negative infinity.
As x approaches positive or negative infinity, the term 1 / (x - 3) tends to zero since the denominator grows much faster than the numerator. Therefore, the horizontal asymptote is y = 1.
In summary:
Domain: All real numbers except x = 3.
Vertical Asymptote: x = 3.
Horizontal Asymptote: y = 1.