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

Given:

The function is:

To find:

The graph of the given function.

Solution:

We have,

It can be written as:

Putting to find the y-intercept.

So, the y-intercept is .

Putting to find the x-intercept.

So, the x-intercept is .

For vertical asymptote, equate the denominator and 0.

So, the vertical asymptote is .

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

So, the horizontal asymptote is .

End behavior of the given function:

as

as

as

as

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

boxplot

the system of inequalities is

step-by-step explanation:

step 1

find the equation of the dashed line

we have the points

(0,0) and (3,1)

the line passes through the origin, so, is a direct variation

find the slope

the equation of the dashed line is

the solution of the inequality a s the shaded area above the dashed line

therefore

the equation of the inequality a is

step 2

find the equation of the solid line

we have the points

(4,0) and (0,4)

find the slope

the equation of the solid line is

the solution of the inequality b s the shaded area below the solid line

therefore

the equation of the inequality b is

step 3

the system of inequalities is