given the function y=2-log(x) find domain, determine vertical asymptote and show work.
log is only defined for positive numbers. So,
domain is x > 0
You know what y = log(x) looks like. Vertical asymptote at x=0.
y = -log(x) is just log(x) reflected about the x-axis. Still has the vertical asymptote at x=0.
y = 2 - log(x) = -log(x) + 2 is just that graph shifted up 2 units. Asymptote does not change.
To find the domain of a function, we need to identify the set of all possible values for the independent variable (x) that yield a meaningful output.
In this case, we have the function y = 2 - log(x). The natural logarithm function (log(x)) is defined only for positive values of x. Therefore, the domain of the given function is all positive real numbers (x > 0).
Next, let's determine the vertical asymptote. A vertical asymptote occurs when the function approaches infinity or negative infinity as x approaches a certain value.
In this case, the vertical asymptote occurs when the logarithmic function approaches negative infinity, since log(x) goes to negative infinity as x approaches zero.
To find the vertical asymptote, we set the argument of the logarithmic function to zero and solve for x:
x = 0
So, the vertical asymptote is x = 0.
To find the domain of a function, we need to find the values of x for which the function is defined. In this case, the given function is y = 2 - log(x).
The function contains the logarithm of x, which is defined only for positive values of x. Thus, the domain of the function is all positive numbers greater than zero.
Domain: x > 0
Now, to determine the vertical asymptote of the function, we need to find the value(s) of x that make the denominator of the function equal to zero. However, in this case, there is no denominator in the function since it only contains a logarithm. Therefore, there is no vertical asymptote for this function.
To summarize:
Domain: x > 0
No vertical asymptote.