The graph of f(x)= (x^2 - 11x + 18)/(x-9) consists of a line and a hole. Find the equation of the line and the coordinates of the hole.
I really have no idea how to do this. I tried factoring it and ended up with (x-2), but I don't know how that helps me. please help
did you notice that
f(x) = (x-9)(x-2)/(x-9) ??
= x-2, x ≠ 9
So for all values of x except x=9
the given function reduces to
f(x) = x-2
which is a straight line.
when x = 9, the original gives you 0/0 which is indeterminate, while the reduced function gives you
(9,7)
So the graph would be the straight line
y = x-2, with a hole at (9,7)
try this:
let x = 8.999 and sub into the original using your calculator, you should get 6.999
let x = 9.00001 in the original
you will get 7.00001
which of course you will get in that exact form if you use y = x-2
However, when you use to calculator to find
f(9) using the original you will get an error , while
f(9) in the simplified version will give you 7
Wow, I didn't realize that finding (x-2) was already finding the equation of the line. Thanks!
To find the equation of the line and the coordinates of the hole in the graph of f(x), you need to consider the factors of the numerator and denominator separately.
First, let's factor the numerator, f(x) = (x^2 - 11x + 18). You correctly factored it as (x-2)(x-9). This means that the numerator has two roots: x = 2 and x = 9.
Next, let's consider the denominator, f(x) = (x-9). This shows that the function has a vertical asymptote at x = 9. At this value, the function is undefined since division by zero is not possible.
Since the factor (x-9) is common to both the numerator and denominator, it creates a hole in the graph at x = 9. To find the y-coordinate of the hole, we can simplify the function by canceling out the common factor:
f(x) = (x^2 - 11x + 18)/(x-9) = (x-2)(x-9)/(x-9)
As you can see, the (x-9) terms in the numerator and denominator cancel each other out, leaving f(x) = x-2. This means that the simplified equation of the line is f(x) = x-2.
To summarize:
- The equation of the line in the graph is f(x) = x-2.
- The graph has a hole at x = 9, indicating that the function is undefined at this point. The y-coordinate of the hole can be found by evaluating f(x) at x = 9, resulting in f(9) = 9-2 = 7. Therefore, the coordinates of the hole are (9, 7).
Remember, when you have a rational function with a common factor in the numerator and the denominator, you can simplify the equation and determine the hole by canceling out the common factor.