Question

How do you find the domain and range of `f(x) = - ( 1 / ( x + 1) )`?

Answer 1

Domain: `" " x < -1 or x > -1`

Range: `" " f(x) < 0 or f(x) > 0`

Using interval notations

Domain:`" "(-oo, -1) uu (-1, oo)`

Range:`" "(-oo, 0) uu (0, oo)`

Explanation:

We are given the function `f(x) = -(1/(x+1))`

To find the Domain of `f(x)`, set `color(green)(x+1 = 0)`

`rArr (x + 1) = 0`

`rArr x = -1`

We must remember that `x = -1` is our Vertical Asymptote

So, our Domain is `(-oo, -1) uu (-1, oo)`

To find the Range of `f(x)`

Let us now observe our original function `f(x) = -(1/(x+1))`

We note that there is no x term in the numerator(NR)

Hence, the degree of numerator is ZERO(0).

The highest degree of the denominator(DR) is ONE(1).

We observe that degree of the DR `>` degree of the NR

Hence, we conclude that `y=0` is our Horizontal Asymptote

Range: `" " f(x) < 0 or f(x) > 0`

Range using interval notations is `(-oo, 0) uu (0, oo)`

Please investigate the graph for a visual comprehension