First, let's consider the function given to us:
[tex]\[ f(x) = -\frac{x-1}{x-1} \][/tex]
Notice that the denominator equals the numerator, so we can simplify the function as follows:
[tex]\[ f(x) = -\frac{x-1}{x-1} = -1 \][/tex]
This simplification is valid for all [tex]\( x \)[/tex] values except [tex]\( x = 1 \)[/tex]. At [tex]\( x = 1 \)[/tex], the function is undefined because the denominator becomes zero.
Understanding that the simplified function [tex]\( f(x) = -1 \)[/tex] is a constant function, we know that:
- A constant function, like [tex]\( f(x) = -1 \)[/tex], does not have a slant (or oblique) asymptote.
A slant asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator in a rational function. However, since here the function simplifies to a constant, the conditions for having a slant asymptote are not satisfied.
Therefore, none of the given options (A, B, C, or D) correctly describe a slant asymptote for this function. The correct conclusion is that there is no slant asymptote for this function.
