Answers to Tutorial on Rational Functions (Part 1)

Detailed answers to the matched exercises in Tutorial on Rational Functions (Part 1) are presented below.

Matched Exercise 1

Find the equation of the rational function \( f \) of the form

\[ f(x) = -\frac{1}{bx + c} \]

whose graph has:

  1. a y-intercept at \( (0, -\frac{1}{4}) \)
  2. a vertical asymptote at \( x = -1 \)

Answer

Using the y-intercept condition:

\[ f(0) = -\frac{1}{c} = -\frac{1}{4} \Rightarrow c = 4 \]

Using the vertical asymptote condition:

\[ bx + c = 0 \Rightarrow x = -\frac{c}{b} = -1 \Rightarrow b = 4 \]

Therefore, the required values are:

\[ b = 4 \quad \text{and} \quad c = 4 \]

Matched Exercise 2

Find the equation of the rational function \( f \) of the form

\[ f(x) = \frac{ax - 2}{bx + c} \]

whose graph has:

  1. an x-intercept at \( (1, 0) \)
  2. a vertical asymptote at \( x = -1 \)
  3. a horizontal asymptote at \( y = 2 \)

Answer

From the x-intercept condition:

\[ ax - 2 = 0 \Rightarrow a(1) - 2 = 0 \Rightarrow a = 2 \]

From the vertical asymptote condition:

\[ bx + c = 0 \Rightarrow -\frac{c}{b} = -1 \Rightarrow c = b \]

From the horizontal asymptote condition:

\[ \frac{a}{b} = 2 \Rightarrow \frac{2}{b} = 2 \Rightarrow b = 1 \]

Since \( c = b \), we have:

\[ c = 1 \]

Therefore, the required values are:

\[ a = 2, \quad b = 1, \quad c = 1 \]