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The solutions to Exercise Set 12 of a calculus course, focusing on finding the domain, vertical and horizontal asymptotes, intercepts, and graphing rational functions. It includes five exercises with detailed step-by-step solutions.
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Exercise 1: Find the domain of the given rational function.
( ) 2 5 4 3
h x x x x
Exercise 2: Use the given graph to complete the statements below.
a) As x โ โโ, f (^) ( x )โ b) As x โ โ2 , โ f (^) ( x )โ c) As x^ โ โ2 ,^ + f^ ( x )โ d) As x โ 2 ,โ f (^) ( x )โ e) As x โ 2 ,+ f (^) ( x )โ f) As x โ โ, f (^) ( x )โ
Exercise 3: Find the vertical and horizontal asymptotes of the given rational function.
( ) (^2)
g x x x x
Exercise 4: Graph the given rational function by finding any symmetry, intercepts, asymptotes, and any additional points.
( )
2 2
r x x x x
Exercise 1: Find the domain of the given rational function.
( ) (^2)
h x x x x
Set the denominator equal to zero and solve for x x 2 + 4x + 3 = 0 (x + 1)(x + 3) = 0 x + 1 = 0 or x + 3 = 0 x = -1 or x = -
Exclude the values that make the denominator zero from the domain
Domain: (^) ( โโ โ, (^3) ) โช โ( 3, โ (^1) ) โช โ( 1, โ)
Exercise 2: Use the given graph to complete the statements below.
a) As x โ โโ, f (^) ( x )โ b) As x โ โ2 , โ f (^) ( x )โ c) As x โ โ2 , + f (^) ( x )โ d) As x โ 2 ,โ f (^) ( x )โ e) As x โ 2 ,+ f (^) ( x )โ f) As x โ โ, f (^) ( x )โ
Exercise 3: Find the vertical and horizontal asymptotes of the given rational function.
g (^) ( x (^) ) = (^) x (^23) โ^^ x + x^7 โ 6
Vertical asymptote Set the denominator equal to zero and solve for x x 2 - x - 6 = 0 (x + 2)(x - 3) = 0 x + 2 = 0 or x - 3 = 0 x = -2 or x = 3 The vertical asymptotes will be at x = -2 and x = 3 Horizontal asymptote Compare degrees of the numerator and denominator degree of numerator: 1 degree of denominator: 2 Since the denominator has a larger degree, the horizontal asymptote will be the x-axis or y = 0.
Exercise 4: Graph the given rational function by finding any symmetry, intercepts, asymptotes, and any additional points.
( )
2 2
r x x x x
Vertical asymptote
x 2 + 4x - 12 = 0 (x + 6)(x - 2) = 0 x + 6 = 0 or x - 2 = 0 x = -6 or x = 2
Horizontal asymptote
degree of numerator: 2 degree of denominator: 2
degrees are the same so the horizontal asymptote will be the ratio of the leading coefficients.
y = 2 1
= 2
Exercise 4 (Continued):
Graph
Exercise 5: Graph the given rational function by finding any symmetry, intercepts, asymptotes, and any additional points.
( )
3 2 r x x^2 x x
Vertical asymptote
x 2 + x = 0 x(x + 1) = 0 x = 0 or x + 1 = 0 x = 0 or x = -
Horizontal asymptote
degree of numerator: 3 degree of denominator: 2
The numerator has the larger degree so there is no horizontal asymptote. However, since the difference in the degrees is 1 there will be a slant asymptote.
Exercise 5 (Continued):
Slant asymptote
Divide the rational function using long division
The quotient of x - 1 is the slant asymptote.
Intercepts
Let x = 0
x cannot be zero since this is the location of one of the vertical asymptotes
Let r(x) = 0 3 2 3 3 3
x x x x x x x
Symmetry
(^3 ) 2 2 3 3 2 2
r x r x x (^) x x x x^ x x x x x x x
r(-x) and r(x) are not the same functions so it is not symmetric about the y-axis.
