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Question Number 78667 by berket last updated on 19/Jan/20

l e t f (x) = (x + 1) \frac{(x + 1) {(x - 3)}^{2}}{{(x - 1)}^{2} (x - 4)} t h e n

a . f i n d x a n d y i n t e r c e p t s

b . f i n d v e r t i c a l a s y m p t o t e a n d h o r i z o n t a l a s y m t o t e

c . f i n d d o m a i n a n d r a n g e o f f

d . d r a w t h e g r a p h o f f

Answered by Rio Michael last updated on 19/Jan/20

a . l e t y = \frac{{(x + 1)}^{2} {(x - 3)}^{2}}{{(x - 1)}^{2} (x - 4)}

y - i n t e r c e p t, x = 0

y = \frac{{(0 + 1)}^{2} {(0 - 3)}^{2}}{{(0 - 1)}^{2} {(0 - 4)}^{2}} = \frac{9}{16}

p o i n t (0, \frac{9}{16})

x - i n t e r c e p t, y = 0

0 = \frac{{(x + 1)}^{2} {(x - 3)}^{2}}{{(x - 1)}^{2} {(x - 4)}^{2}}

{[(x + 1) (x - 3)]}^{2} = 0

x = - 1 o r x = 3

p o i n t (- 1, 0) a n d (3, 0)

b . V e r t i c a l a s y m p t o t e s {(x - 1)}^{2} (x - 4) = 0

e i t h e r {(x - 1)}^{2} = 0 \Rightarrow x = 1 \lor x = 4

H o r i z o n t a l a s y m p t o t e

\lim_{x \to \infty} f (x) = \infty h e n c e n o h o r i z o n t a l a s y m p t o t e .

c . d o m a i n = v e r t i c a l a s y m p o t e s