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points of discontinuity of rational function y= x²+2 x / x²-7 x-18 is x = 9 & x = -2.
How to find the points of discontinuity of the rational function & find the x - and y -intercepts of the rational function?
points of discontinuity:
In rational functions, points of discontinuity are defined as fractions with undefined or zero as their denominator. When the denominator of a fraction is 0, the fraction is no longer defined and shows up as a hole or break in the graph.
points of discontinuity removable:
A point on the graph that is removable discontinuity is one that may be filled in with a single value despite not existing.If an is zero for a factor in the denominator that shares a factor with a factor in the numerator, a removable discontinuity at x=a in the graph of a rational function occurs.
points of discontinuity non removable:
A function's break that cannot be repaired with a single point is referred to as a non-removable discontinuity. It cannot be filled in with a single point, a vertical asymptote (a vertical line that the graph approaches but never crosses because the function is undefined at that x-value) is non removable.
given that
[tex]y = \frac{ {x}^{2} + 2x}{ {x}^{2} - 7x - 18 } [/tex]
now take the denominator and find the factors of it.
[tex] {x}^{2} - 7x - 18 = 0[/tex]
[tex] {x}^{2} - 9x + 2x - 18 = 0[/tex]
x (x -9) + 2 (x - 9) = 0
(x - 9) (x + 2) = 0
[tex]y = \frac{ {x}^{2} + 2x}{(x - 9)(x + 2)} [/tex]
now, take common on numerator
[tex]y = \frac{2(x + 2)}{(x - 9)(x + 2)} [/tex]
cancel the common factor in both numerator and denominator
[tex]y = \frac{x}{x - 9} [/tex]
now find the points of discontinuity.
x-9=0 , x+2=0
x = +9 , x = -2
now, x = +9 and x = -2
Hence the points of discontinuity of rational function y= x²+2 x / x²-7 x-18 is x = 9 & x = -2.
Learn more about rational functions from here:
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