Question

Which graph represents the function f (x) = StartFraction 1 Over x + 3 EndFraction minus 2?

Answer 1

The Answer Is A.

Explanation:

Answer 2

Given:

The function is:

`f(x)=(1)/(x+3)-2`

To find:

The graph of the given function.

Solution:

We have,

`f(x)=(1)/(x+3)-2`

It can be written as:

`f(x)=(1-2(x+3))/(x+3)`

`f(x)=(1-2x-6)/(x+3)`

`f(x)=(-2x-5)/(x+3)`

Putting
`x=0` to find the y-intercept.

`f(0)=(-2(0)-5)/((0)+3)`

`f(0)=(-5)/(3)`

So, the y-intercept is
`(-5)/(3)`.

Putting
`f(x)=0` to find the x-intercept.

`0=(-2x-5)/(x+3)`

`0=-2x-5`

`2x=-5`

`x=(-5)/(2)`

`x=-2.5`

So, the x-intercept is
`-2.5`.

For vertical asymptote, equate the denominator and 0.

`x+3=0`

`x=-3`

So, the vertical asymptote is
`x=-3`.

The degrees of numerator and denominator are equal, so the horizontal asymptote is the ratio of leading coefficients.

`y=(-2)/(1)`

`y=-2`

So, the horizontal asymptote is
`y=-2`.

End behavior of the given function:

`f(x)\to -2` as
`x\to -\infty`

`f(x)\to -\infty` as
`x\to -3^-`

`f(x)\to \infty` as
`x\to -3^+`

`f(x)\to -2` as
`x\to \infty`

Using all these key features, draw the graph of given function as shown below.