Question

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

Answer 1

Given:

The function is:

$f(x)=\dfrac{1}{x+3}-2$

To find:

The graph of the given function.

Solution:

We have,

$f(x)=\dfrac{1}{x+3}-2$

It can be written as:

$f(x)=\dfrac{1-2(x+3)}{x+3}$

$f(x)=\dfrac{1-2x-6}{x+3}$

$f(x)=\dfrac{-2x-5}{x+3}$

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

$f(0)=\dfrac{-2(0)-5}{(0)+3}$

$f(0)=\dfrac{-5}{3}$

So, the y-intercept is $\dfrac{-5}{3}$.

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

$0=\dfrac{-2x-5}{x+3}$

$0=-2x-5$

$2x=-5$

$x=\dfrac{-5}{2}$

$x=-2.5$

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.

$y=\dfrac{-2}{1}$

$y=-2$

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.