14)  Consider the function whose rule is
Solution:  
 f is a rational function.
Its numerator has no zeros and therefore the function f has no zeros. It follows that the graph of the function f will have no x-intercepts.

The zero of the denominator is 3.  Therfore the graph of the function f will have a vertical asympotote at x = 3.

It also follows that the domain of f is all real numbers other than 3.

The denominator has degree 1 and the numerator has degree 0. Because the degree of the denominator is greater than the degree of the numerator, the x-axis is a horizontal asymptote of the graph of the function f.

Because the numerator 1 is positive for all values of x in the domain, f(x) > 0 when x - 3 >0 and f(x) < 0 when x - 3 < 0.
Therefore f(x) > 0 if x > 3 and f(x) < 0 if x < 3.

Therefore the graph of the function f is above the x-axis for all values of x > 3 and it is below the x-axis for all values of x < 3.

                                

18)  Consider the function whose rule is
Solution:  
 P is a rational function.
The zeros of the numerator is 1/3.  This zero is the zero of the function P. Since this zero is a real number, it represents an x-intercept.

The zero of the denominator is 1.  Since this zero is a real number, it represents a vertical asymptote of the graph of the function P.

It also follows that the domain of the function P is all real numbers other than 1.

The degree of both the numerator and denominator is 1. Since numerator and denominator have the same degree, the function P has a horizontal asymptote at y = (-3/-1) = 3.

Observe that
1)     P(x) > 0 when both the numerator and denominator are positive or when both the numerator and denominator are negative.
2)     P(x) < 0 when one of the numerator and denominator is positive and the other negative.
Look at the sketch of the numerator and the denominator below to determine where P(x) > 0 and where P(x) < 0.
                             
Clearly both the numerator and denominator are above the x-axis (so are positive) when x < 1/3. Then the quotient P(x) >0
Clearly both the numerator and denominator are below the x-axis (so are negative) when x > 1. Then the quotient P(x) >0
Clearly between 1/3 and 1 the numerator is negative (below the x-axis) and the denominator is positive (above the x-axis). The the quotient P(x) < 0. The skecth below shows the excluded regions of the plane.
                         

It is possible for the graph of a function to cross its horizontal asymptotes. Therefore it is wise to determine if that happens. The graph of the function P crosses the horizontal asymptote y = 3 if and only
The graph of the function P does not cross its horizontal asymptote y =3

if P(x ) = 3 which we have determined earlier to occurr at x = 0. So the graph of P crosses its horizontal asymptote at the origin.

The graph of the rational function P is shown below with the horizontal and vertical asymptotes drawn in for reference.

                   

 

24)  Consider the function whose rule is

Solution:  
g is a rational function.
The zero of the numerator is 0.  This zero is the zero of the function g. Since this zero is a real number, it represents an x-intercept.

The zeros of the denominator are 3 and -3.  Since these zeros are real numbers, they represent vertical asymptotes of the graph of the function g.

It also follows that the domain of the function g is all real numbers other than 3 and -3.

The denominator has degree 2 and the numerator has degree 1. Because the degree of the denominator is greater than the degree of the numerator, the x-axis is a horizontal asymptote of the graph of the function f.

Observe that
1)     g(x) > 0 when both the numerator and denominator are positive or when both the numerator and denominator are negative.
2)     g(x) < 0 when one of the numerator and denominator is positive and the other negative.
Look at the sketch of the numerator and the denominator below to determine where g(x) > 0 and where g(x) < 0.

Notice that the vertical red lines drawn at the asymptotes and the zero of the function divide the coordinate plane into 4 vertical strips. We can describe these strips as x < -3,     -3 < x < 0,     0< x < 3,       3 < x

In the strip x < -3, the graph of the numerator is below the x-axis (negative) and the graph of the denominator is above the x-axis (positive) . Therefore in the strip x < -3, g(x) will be negative (below the x-axis). The half-strip above the x-axis and to the left of x = -3 is an excluded region.

In the strip -3< x < 0, the graph of both numerator and denominator are below (negative) the x-axis. Therefore in the strip -3< x < 0, g(x) will be positive (above the x-axis). The half-strip below the x-axis and between the lines x = -3 and x = 0 is an excluded region.

In the strip 0 < x < 3, the graph of both numerator is above (positive) the x-axis and denominator is below (negative) the x-axis. Therefore in the strip 0 < x < 3, g(x) will be negative (below the x-axis). The half-strip above the x-axis and between the lines x = 0 and x = 3 is an excluded region.

In the strip 3< x, the graph of both numerator and denominator are above (positive) the x-axis. Therefore in the strip 3< x, g(x) will be positive (above the x-axis). The half-strip below the x-axis and to the right of the line x = 3 is an excluded region.

This analysis is summarized below by shading excluded regions.

 

It is possible for the graph of a function to cross its horizontal asymptotes. Therefore it is wise to determine if that happens. The graph of the function g crosses the horizontal asymptote y = 0 if and only if g(x ) = 0 which we have determined earlier to occurr at x = 0. So the graph of g crosses its horizontal asymptote at the origin.

The graph of the rational function g is shown below with the vertical asymptotes drawn in for reference.   

 

Finally the graph of the function g may be sketched as shown below.

One final graphic shows how the graph of the function g avoids the excluded regions of the plane.

                                            

 

32) Consider the function whose rule is
Solution:  
To help with the analysis it is helpful to write
 f is a rational function.
The zeros of the numerator are the complex numbers i and -i.  These zeros are zeros of the function f. Since these zeros are complex numbers, they do not represent x-intercepts.

The zeros of the denominator are 3 and -5.  Since these zeros are real numbers, they represent vertical asymptotes of the graph of the function f.

It also follows that the domain of the function f is all real numbers other than 3 and -5.

The degree of both the numerator and denominator is 2. Since numerator and denominator have the same degree, the function f has a horizontal asymptote at y = (3/1) = 3.

Observe that
1)     f(x) > 0 when both the numerator and denominator are positive or when both the numerator and denominator are negative.
2)     f(x) < 0 when one of the numerator and denominator is positive and the other negative.

The numerator of the function f is positive for all values of x. Therefore the function f will be positive when its denominator is positive and will be negative when its denominator is negative.
The denominator of f is a quadratic whose graph is a parabola which opens up. Therefore it is below (negative) the x-axis between -5 and 3 (its zeros) and is above (positive) the x-axis everywhere else.
This analysis is summarized below by shading excluded regions.

It is possible for the graph of a function to cross its horizontal asymptotes. Therefore it is wise to determine if that happens. The graph of the function f crosses the horizontal asymptote y = 3 if and only if

Therefore the graph of f crosses the asymptote y = 3 at x = 8.   The point of intersection is (8, 3)

The graph of the rational function f is shown below with the horizontal and vertical asymptotes drawn in for reference.