Identify the rational function with smallest degree in the numerator and denominator whose graph matches the graph given below. The function has an x-intercept at x=3 and a y-intercept at y=−3/4.

Step-by-step explanation:

At the x-intercept

x

=

−

1

corresponding to the

(

x

+

1

)

2

factor of the numerator, the graph bounces, consistent with the quadratic nature of the factor.

At the x-intercept

x

=

3

corresponding to the

(

x

−

3

)

factor of the numerator, the graph passes through the axis as we would expect from a linear factor.

At the vertical asymptote

x

=

−

3

corresponding to the

(

x

+

3

)

2

factor of the denominator, the graph heads towards positive infinity on both sides of the asymptote, consistent with the behavior of the function

f

(

x

)

=

1

x

2

.

At the vertical asymptote

x

=

2

, corresponding to the

(

x

−

2

)

factor of the denominator, the graph heads towards positive infinity on the left side of the asymptote and towards negative infinity on the right side, consistent with the behavior of the function

f

(

x

)

=

1

x

23

step-by-step explanation:

answer

the correct answer is

explanation

the given function is

the excluded values are obtained by solving:

we factor to obtain:

the excluded values are