Consider the given function.
f(x)=x^2-5x-14/x-7
Recall that division by zero is not defined. Therefore, the rational function is undefined where x-7=0.
x-7=0 ⇔ x=7
This means that x=7 is not included in the domain.
Domain
All real numbers except x=7
Let's consider the given function.
f(x)=x^2-5x-14/x-7
Notice that the numerator can be factored. Let's factor this expression and cancel out any common factors between the numerator and the denominator.
We canceled out the factor x-7. Therefore, there is a hole in the graph at x=7. Since there are no other factors in the denominator, there are no vertical asymptotes.
Horizontal and Oblique Asymptotes
To find the horizontal and oblique asymptotes, we can use a set of rules. To properly use these rules, in the following table, m and n are the degree of the numerator and denominator, respectively.
f(x)=ax^m/bx^n
Asymptote
Asymptote Type
m
y=0
Horizontal
m>n
none
-
m=n
y=a/b
Horizontal
m-n=1
y= the quotient of the polynomials with no remainder
Oblique
Now, consider the function one more time.
Let's look at the degrees of the numerator and denominator for our function.
f(x)=x^2-5x-14/x^1-7
We can see that the degree of the numerator is higher than the degree of the denominator. Therefore, there are no asymptotes.
Intercepts
The intercepts of the function are the points at which the graph intersects the axes.
x-intercepts
The x-intercepts are the points where the graph intersects the x-axis. At these points, the value of the y-coordinate is zero. Let's substitute 0 for f(x) in the given function and solve for x.
By the Zero Product Property, the equation equals 0 when either of the factors is equal to 0.
c|c
x-7=0 & x+2=0
⇕ & ⇕
x=7 & x=-2
We found that if f(x)=0, the value of x is 7 or -2. However, x=7 is not included in the domain of the function. Therefore, the only x-intercept occurs at the point (-2,0).
y-intercept
The y-intercept is the point where the graph intersects the y-axis. At this point, the value of the x-coordinate is zero.
Let's substitute 0 for x in the given function and simplify.
