find-the-set-of-values-of-x-for-which-y-is-real-if-y-x-2-x-1-x-2-x-2-x-R-

Question Number 63534 by Rio Michael last updated on 05/Jul/19

f i n d t h e s e t o f v a l u e s o f x f o r w h i c h y i s r e a l i f

y = \frac{(x - 2) (x - 1)}{x + 2}, x \neq - 2, x \in R

Answered by MJS last updated on 05/Jul/19

the answer is already given : x \in R ∖ {- 2}

Commented by Rio Michael last updated on 05/Jul/19

w h y i s i t s o ?

Commented by Prithwish sen last updated on 05/Jul/19

for x = - 2 y is undefined

Commented by MJS last updated on 05/Jul/19

\forall x \in R : (x - 2) (x - 1) \in R

\forall t \in R : \frac{t}{0} is not defined

\forall t \in R : \frac{t}{x + 2} \in R \Leftrightarrow x + 2 \neq 0 \Leftrightarrow x \neq - 2

\forall x \in R ∖ {- 2} : \frac{(x - 2) (x - 1)}{x + 2} \in R

Commented by Rio Michael last updated on 11/Jul/19

s o t h e v a l u e o f x f o r w h i c h y i s r e a l i s t h e v a l u e o f x f o r w h i c h

y i s u n d e f i n e d ?

Commented by MJS last updated on 12/Jul/19

\forall x \in R ∖ {- 2} : \frac{(x - 2) (x - 1)}{x + 2} \in R means

\forall for all

x \in R ∖ {- 2} x real but x \neq - 2

: the following is true

\frac{(x - 2) (x - 1)}{x + 2} \in R this expression is real

Commented by Rio Michael last updated on 12/Jul/19

i g e t i t n o w .

Commented by Rio Michael last updated on 12/Jul/19

m e a n s i f g i v e n s o m e t h i n g l i k e

y = (x + 3) (x + 6) t o f i n d t h e v a l u e o f x f o r w h i c h y i s r e a l

x \in R ?

Commented by MJS last updated on 12/Jul/19

yes

y = (x - 3) (x + 5)

x \in R \Rightarrow y \in R

y = \frac{1}{(x - 3) (x + 5)}

x \in R ∖ {- 5, 3} because y is not defined for

these values

y = \frac{x - 3}{x + 5}

x \in R ∖ {- 5}

Commented by Rio Michael last updated on 12/Jul/19

t h a n k s s o m u c h