Answers:
- root: x = 0
- Vertical asymptotes: x = 5 and x = -5
- Horizontal asymptote: y = 0
The term "root" is another way of saying "zero of a function"
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Explanation:
To find the roots, we need to solve f(x) = 0 for x
f(x) = 0
5x/(x^2 - 25) = 0
5x = 0*(x^2-25)
5x = 0
x = 0/5
x = 0
Plugging x = 0 into f(x) leads to f(x) = 0.
The root is x = 0 meaning the x intercept is at 0 on the x axis number line. This is the origin (0,0).
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To find the vertical asymptotes, set the denominator equal to 0 and solve for x
x^2 - 25 = 0
x^2 - 5^2 = 0
(x-5)(x+5) = 0 ... difference of squares rule
x-5 = 0 or x+5 = 0
x = 5 or x = -5
If either x = 5 or x = -5, then then the denominator x^2-25 is zero
We cannot divide by zero, so these values are excluded from the domain, and this produces the vertical asymptotes.
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The horizontal asymptote is y = 0 because the degree of the numerator is 1, while the degree of the denominator is 2. The denominator's degree is larger, which leads to y = 0. The rules for finding the horizontal asymptote are
- If m < n, then the horizontal asymptote is y = 0
- If m = n, then the horizontal asymptote is y = a/b
- If m > n, then there is no horizontal asymptote (use polynomial long division to find the oblique asymptote)
For those rules above, m and n are the degrees of the numerator and denominator respectively. The a,b refers to the leading coefficients of the numerator and denominator.
