Suppose that h(x) = x^2 - 9/x^2 - 4. Find the following information about h(x).

Question

Suppose that h(x) = x^2 - 9/x^2 - 4. Find the following information about h(x). Domain of h(x): y-intercept: x-intercept(s): Is h(x) even or odd? Vertical asymptote(s): Horizontal asymptote(s): h'(x) = Interval (s) h(x) is increasing: Interval(s) h(x) is decreasing: Local max(s) (x,y): Local min(s) (x,y): h"(x) = Interval(s) h(x) is concave up: Interval (s) h(x) is concave down: Inflection point(s) (x.y): Sketch y -h(x) on the graph below. Suppose that g(x) = 2 Find the following information about g(x). Domain of g(x): y-intercept.: x-intcrcopt(s): Is g(x) even or odd? Vertical asymptote(s): Horizontal asymptote(s): g'(x) = G. Interval(s) g{x) is increasing: Interval(s) g(x) is decreasing: Local max(s) (x, y): Local min(s) Interval(s) g(x) is concave up: Interval (s) g(x) is concave down: K. Inflection point.(s) (x. y): Sketch y = g(x) on the graph below.

Explanation / Answer

given h(x)=y =(x2-9)/(x2-4)

A) domain :

function is defined when x2-40

=>x-2,2

domain is (-,-2)U(-2,2)U(2,)

B) for y intrcept x =0

=>h(0) =9/4

y intercept (x,y)=(0,9/4)

for x intercept y =0

=>(x2-9)/(x2-4)=0

=>x =-3, x =3

x intercepts are (-3,0)and (3,0)

C)h(-x)=((-x)2-9)/((-x)2-4)

=(x2-9)/(x2-4)

=h(x)

h(x) is even

D) veritcal asymptotes

denominator =0

=>x2-4=0

=>x =2, x =-2 are vertical aymptotes

E) horizontal asymptotes

y =limx-> h(x)

y =limx-> (x2-9)/(x2-4)

y =1 is horizontal aymptote

y =limx->- h(x)

y =limx->- (x2-9)/(x2-4)

y =1 is horizontal aymptote

F)h(x)=(x2-9)/(x2-4)

differentiate with respect to x, quotient rule:(u/v)'=(u'v -uv')/v2

h'(x)=[(2x-0)(x2-4) -(x2-9)(2x-0)]/(x2-4)2

h'(x)=[2x3-8x -2x3+18x]/(x2-4)2

h'(x)=10x/(x2-4)2

G) increasing when h'(x)>0

=>10x/(x2-4)2>0

=>x=(0,2)U(2,)

decreasing when h'(x)<0

=>10x/(x2-4)2<0

=>x=(-,-2)U(-2,0)

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