Answers

  • Brainly User
2016-01-09T19:40:48+08:00
(x + 1) y = 1

Solution:
Step 1:  Isolate y to -re-write the expression as function, y = f(x)

 \frac{(x+1)y}{x+1} =  \frac{1}{x+1}

f(x) =  \frac{1}{x+1}

Step 2: Solve for domain.

x + 1 = 0
x - 1 = 0 - 1
x = -1

Since x + 1 is the denominator of  \frac{1}{x+1} , when -1 is substituted to denominator x + 1 ⇒  (-1) + 1 = 0.  

Note that in any rational expression, denominator can not be 0, because it will render the expression 'undefined".

Therefore:
Domain = {x/x ≠  -1)
              =  x < -1,   x > -1   
              = {-2, -3, -4,...}  and  {0, 1, 2, 3, ...}

Interval Notation   (⁻∞, -1) U (-1, ⁺∞)

Step 3:  Solve for range
Remember that the range is the combined domain of its inverse functions.

f⁻¹(x) = -1 + 1/x
f⁻¹(x) =  \frac{1-x}{x}
Find the value of denominator x:
x = 0

Again, if the denominator is equal to 0, the rational expression is undefined.
Therefore:

Range = {y/y ≠0}
            =  y < 0,  y > 0    
            = {-1, -2, -3, ...}  and  {1, 2, 3, ...}

Interval Notation:   (⁻∞, 0)  U  (0, ⁺∞)
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