How do you graph #f(x)=(x+3)/((x+1)(x-3))# using holes, vertical and horizontal asymptotes, x and y intercepts?

Question

Julia Adams · Accepted Answer

see explanation.

#color(blue)"Asymptotes"# The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes. #"solve " (x+1)(x-3)=0rArrx=-1" and " x=3# #rArrx=-1" and " x=3" are the asymptotes"# Horizontal asymptotes occur as #lim_(xto+-oo),f(x)toc" ( a constant)"# divide terms on numerator/denominator by the highest power of x, that is #x^2# #f(x)=(x/x^2+3/x^2)/(x^2/x^2-(2x)/x^2-3/x^2)=(1/x+3/x^2)/(1-2/x-3/x^2# as #xto+-oo,f(x)to(0+0)/(1-0-0)# #rArry=0" is the asymptote"# Holes occur when there is a duplicate factor on the numerator/denominator. This is not the case here hence there are no holes. #color(blue)"Intercepts"# #x=0toy=3/(-3)=-1larrcolor(red)" y-intercept"# #y=0tox+3=0tox=-3larrcolor(red)" x-intercept"# graph{(x+3)/(x^2-2x-3) [-10, 10, -5, 5]}

Eva Adamson · Answer

undefined To graph the function f(x) = (x+3)/((x+1)(x-3), we can analyze its holes, vertical and horizontal asymptotes, and x and y intercepts.

Holes: To find any holes in the graph, we need to identify values of x that make the denominator zero. In this case, the denominator (x+1)(x-3) will be zero when x = -1 and x = 3. These values create holes in the graph.

Vertical asymptotes: Vertical asymptotes occur when the denominator of a rational function is zero, but the numerator is not zero. In this case, the vertical asymptotes will be x = -1 and x = 3.

Horizontal asymptotes: To determine the horizontal asymptotes, we compare the degrees of the numerator and denominator. In this case, the degree of the numerator is 1, and the degree of the denominator is 2. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote will be y = 0.

X-intercepts: To find the x-intercepts, we set the numerator equal to zero and solve for x. In this case, (x+3) = 0, so x = -3. Therefore, the x-intercept is (-3, 0).

Y-intercept: To find the y-intercept, we substitute x = 0 into the function. In this case, f(0) = (0+3)/((0+1)(0-3)) = 3/(-3) = -1. Therefore, the y-intercept is (0, -1).

By considering these aspects, we can graph the function f(x) = (x+3)/((x+1)(x-3)) accurately.