Answer:
The answer is explained below
Step-by-step explanation:
The equation f(x) = [tex]\frac{x-2}{x-4}[/tex]
1) Points of discontinuity
To find the Points of discontinuity, we check if there is a common factor between the numerator and denominator.
From The equation f(x) = [tex]\frac{x-2}{x-4}[/tex] there is no common factor between the numerator and denominator therefore there is no point of discontinuity
2) holes
To find the holes, we check if there is a common factor between the numerator and denominator.
From The equation f(x) = [tex]\frac{x-2}{x-4}[/tex] there is no common factor between the numerator and denominator therefore there is no hole
3) Vertical asymptotes
To find the Vertical asymptotes, we equate the denominator to be equal to zero. Therefore from The equation f(x) = [tex]\frac{x-2}{x-4}[/tex], the denominator is:
x - 4 = 0
x = 4
The Vertical asymptotes is at x = 4
4) x-intercepts
To find x intercept, equate the function to zero.
f(x) = [tex]\frac{x-2}{x-4}[/tex]
0 = [tex]\frac{x-2}{x-4}[/tex]
x - 2 = 0
x = 2
The x intercept is at x = 2
5) horizontal asymptote
If the denominator has a higher degree than the numerator, the horizontal asymptote is at y = 0. If the denominator has the same degree as the numerator, the horizontal asymptote is at y = [tex]\frac{a}{b}[/tex], where a and b are the leading coefficients of the numerator and denominator.
For The equation f(x) = [tex]\frac{x-2}{x-4}[/tex], If the denominator has the same degree as the numerator. Therefore the horizontal asymptote is at f(x) = [tex]\frac{x}{x}=1[/tex], the horizontal asymptote is at y = 1
