Question

test (topics 1.7 - 1.11) rational functions
name:
section 1: no calculators allowed
directions: read the following problems carefully. work out each question and circle the best answer. be sure to write
your answer choices on the separate answer sheet as well.

1. which of the following rational functions have a horizontal asymptote?

i. ( f(x)=\frac{3 x^{2}+x - 1}{x^{2}+4} ) ii. ( g(x)=\frac{(x + 1)(2 x - 3)^{2}}{(5 x - 7)^{3}} ) iii. ( h(x)=\frac{x - 4}{x^{3}+x^{2}-6} )
(a) i only (b) i and ii only (c) i and iii only (d) i, ii, and iii

1. let ( k ) be the rational function defined by ( k(x)=\frac{(x - 3)^{4}(x + 2)^{5}}{(x - 3)^{5}(x + 2)^{4}} ). which of the following statements about the

graph of ( k ) is correct?
(a) the graph of ( k ) has a vertical asymptote at ( x=-2 ) and a hole at ( x = 3 ).
(b) the graph of ( k ) has a vertical asymptote at ( x = 3 ) and a hole at ( x=-2 ).
(c) the graph of ( k ) has vertical asymptotes at ( x=-2 ) and ( x = 3 ).
(d) the graph of ( k ) has holes at ( x=-2 ) and ( x = 3 ).

1. the graph of the rational function ( h ) has a horizontal asymptote of ( y=-3 ). which of the following could be an

cxpression for ( h(x) )?
(a) ( \frac{x^{2}-4 x + 7}{-3 x^{2}+5} )
(b) ( \frac{3 x^{2}-5 x - 2}{x^{2}-x + 1} )
(c) ( \frac{-3 x^{2}+x - 2}{x^{2}+5 x - 6} )
(d) ( \frac{-6 x^{3}+7 x^{2}-12}{2 x^{2}+11} )

Explanation:

Step1: Horizontal Asymptote Rules

For a rational function \(y = \frac{f(x)}{g(x)}=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}\), if \(n\lt m\), \(y = 0\); if \(n=m\), \(y=\frac{a_n}{b_m}\); if \(n\gt m\), no horizontal asymptote.

• For \(f(x)=\frac{3x^2 + x-1}{x^2+4}\), \(n = m=2\), \(y=\frac{3}{1}=3\) (horizontal asymptote exists).
• For \(g(x)=\frac{(x + 1)(2x-3)^2}{(5x-7)^3}\), \(n=3\), \(m = 3\), \(y=\frac{1\times2^2}{5^3}=\frac{4}{125}\) (horizontal asymptote exists).
• For \(h(x)=\frac{x - 4}{x^3+x^2-6}\), \(n=1\), \(m=3\), \(y = 0\) (horizontal asymptote exists).

Step2: Holes and Vertical Asymptotes

For \(k(x)=\frac{(x - 3)^2(x + 2)}{(x - 3)^3(x + 2)^2}\), simplify to \(k(x)=\frac{1}{(x - 3)(x + 2)}\) (\(x
eq3\) and \(x
eq - 2\)). When a factor cancels out (e.g., \((x - 3)\) and \((x + 2)\) in numerator and denominator), there are holes at \(x=3\) and \(x=-2\) (since they make the original function undefined before simplification).

Step3: Horizontal Asymptote for \(h(x)\)

• For \(y=\frac{-6x^3+7x^2-12}{2x^2 + 11}\), \(n=3\), \(m=2\), no horizontal asymptote.
• For \(y=\frac{x^2-4x + 7}{-3x^2+5}\), \(n=m = 2\), \(y=\frac{1}{-3}=-\frac{1}{3}\).
• For \(y=\frac{3x^2-5x-2}{x^2-x + 1}\), \(n=m=2\), \(y = 3\).
• For \(y=\frac{-3x^2+x-2}{x^2+5x-6}\), \(n=m=2\), \(y=\frac{-3}{1}=-3\).

Answer:

1. (D)
2. (D)
3. (C)