Asymptote Hunters Worksheet
ASYMPTOTE HUNTERS
Level: Rational Function Mapping
Engineer:
Date:
Identify the domain and range for each rational function provided below using Set-Builder Notation.
Target 01
Which of the following represents the domain and range of the function \[ f(x) = \frac{4}{x} \]
A: Domain: \( \{x | x \in \mathbb{R}, x \neq 4\} \); Range: \( \{f(x) | f(x) \in \mathbb{R}, f(x) \neq 0\} \)
B: Domain: \( \{x | x \in \mathbb{R}, x \neq 0\} \); Range: \( \{f(x) | f(x) \in \mathbb{R}, f(x) \neq 0\} \)
C: Domain: \( \{x | x \in \mathbb{R}, x \neq 0\} \); Range: \( \{f(x) | f(x) \in \mathbb{R}, f(x) \neq 4\} \)
D: Domain: \( \{x | x \in \mathbb{R}\} \); Range: \( \{f(x) | f(x) \in \mathbb{R}\} \)
Target 02
Which of the following represents the domain and range of the function \[ g(x) = \frac{1}{x-3} + 2 \]
A: Domain: \( \{x | x \neq -3\} \); Range: \( \{g(x) | g(x) \neq 2\} \)
B: Domain: \( \{x | x \neq 3\} \); Range: \( \{g(x) | g(x) \neq -2\} \)
C: Domain: \( \{x | x \neq 3\} \); Range: \( \{g(x) | g(x) \neq 2\} \)
D: Domain: \( \{x | x \neq 2\} \); Range: \( \{g(x) | g(x) \neq 3\} \)
Target 03
Which of the following represents the domain and range of the function \[ h(x) = \frac{-5}{2x+6} - 1 \]
A: Domain: \( \{x | x \neq -3\} \); Range: \( \{h(x) | h(x) \neq -1\} \)
B: Domain: \( \{x | x \neq 3\} \); Range: \( \{h(x) | h(x) \neq -1\} \)
C: Domain: \( \{x | x \neq -3\} \); Range: \( \{h(x) | h(x) \neq 1\} \)
D: Domain: \( \{x | x \neq -6\} \); Range: \( \{h(x) | h(x) \neq -1\} \)
Target 04
Which of the following represents the domain and range of the function \[ f(x) = \frac{x-4}{x+5} \]
A: Domain: \( \{x | x \neq 4\} \); Range: \( \{f(x) | f(x) \neq -5\} \)
B: Domain: \( \{x | x \neq -5\} \); Range: \( \{f(x) | f(x) \neq -4\} \)
C: Domain: \( \{x | x \neq -5\} \); Range: \( \{f(x) | f(x) \neq 1\} \)
D: Domain: \( \{x | x \neq 5\} \); Range: \( \{f(x) | f(x) \neq -1\} \)
Target 05
Which of the following represents the domain and range of the function \[ g(x) = \frac{2x+6}{x-1} \]
A: Domain: \( \{x | x \neq 1\} \); Range: \( \{g(x) | g(x) \neq 2\} \)
B: Domain: \( \{x | x \neq -1\} \); Range: \( \{g(x) | g(x) \neq 2\} \)
C: Domain: \( \{x | x \neq 1\} \); Range: \( \{g(x) | g(x) \neq 6\} \)
D: Domain: \( \{x | x \neq -6\} \); Range: \( \{g(x) | g(x) \neq -1\} \)
Target 06
Which of the following represents the domain and range of the function \[ h(x) = \frac{3x-9}{x+2} \]
A: Domain: \( \{x | x \neq 2\} \); Range: \( \{h(x) | h(x) \neq 3\} \)
B: Domain: \( \{x | x \neq -2\} \); Range: \( \{h(x) | h(x) \neq 3\} \)
C: Domain: \( \{x | x \neq -2\} \); Range: \( \{h(x) | h(x) \neq -9\} \)
D: Domain: \( \{x | x \neq 9\} \); Range: \( \{h(x) | h(x) \neq -2\} \)
Target 07
Which of the following represents the domain and range of the function \[ f(x) = \frac{10}{(x-4)^2} + 5 \]
A: Domain: \( \{x | x \neq 4\} \); Range: \( \{f(x) | f(x) > 5\} \)
B: Domain: \( \{x | x \neq -4\} \); Range: \( \{f(x) | f(x) > 5\} \)
C: Domain: \( \{x | x \neq 4\} \); Range: \( \{f(x) | f(x) \neq 5\} \)
D: Domain: \( \{x | x \neq 5\} \); Range: \( \{f(x) | f(x) > 4\} \)
Target 08
Which of the following represents the domain and range of the function \[ g(x) = \frac{-2}{x+8} - 4 \]
A: Domain: \( \{x | x \neq 8\} \); Range: \( \{g(x) | g(x) \neq -4\} \)
B: Domain: \( \{x | x \neq -8\} \); Range: \( \{g(x) | g(x) \neq -4\} \)
C: Domain: \( \{x | x \neq -8\} \); Range: \( \{g(x) | g(x) \neq 4\} \)
D: Domain: \( \{x | x \neq -4\} \); Range: \( \{g(x) | g(x) \neq -8\} \)
Target 09
Which of the following represents the domain and range of the function \[ h(x) = \frac{4x}{x-2} \]
A: Domain: \( \{x | x \neq 2\} \); Range: \( \{h(x) | h(x) \neq 4\} \)
B: Domain: \( \{x | x \neq -2\} \); Range: \( \{h(x) | h(x) \neq 4\} \)
C: Domain: \( \{x | x \neq 2\} \); Range: \( \{h(x) | h(x) \neq 0\} \)
D: Domain: \( \{x | x \neq 4\} \); Range: \( \{h(x) | h(x) \neq 2\} \)
Target 10
Which of the following represents the domain and range of the function \[ f(x) = \frac{1}{x+10} + 7 \]
A: Domain: \( \{x | x \neq 10\} \); Range: \( \{f(x) | f(x) \neq 7\} \)
B: Domain: \( \{x | x \neq -10\} \); Range: \( \{f(x) | f(x) \neq -7\} \)
C: Domain: \( \{x | x \neq -7\} \); Range: \( \{f(x) | f(x) \neq -10\} \)
D: Domain: \( \{x | x \neq -10\} \); Range: \( \{f(x) | f(x) \neq 7\} \)
© 2026 Asymptote Analysis Bureau • Mission: Zero Divide
Asymptote Hunters Teacher Edition
TEACHER'S EDITION
Asymptote Hunters Answer Key
Confidential Instructional Guide
Quick Response Key
Target 01 B
Target 02 C
Target 03 A
Target 04 C
Target 05 A
Target 06 B
Target 07 A
Target 08 B
Target 09 A
Target 10 D
Solution Rationale
T01-T03: Transformation Form \( y = \frac{a}{x-h} + k \)
Identify the Vertical Asymptote (VA) at \( x = h \) and Horizontal Asymptote (HA) at \( y = k \). The domain excludes \( h \) and the range excludes \( k \) in set notation.
T04-T06, T09: Rational Form \( y = \frac{ax+b}{cx+d} \)
VA: Solve \( cx+d = 0 \). HA: Ratio of leading coefficients \( y = a/c \). Domain excludes VA, Range excludes HA.
T07: Squared Denominator \( y = \frac{a}{(x-h)^2} + k \)
VA at \( x = h \). Since the squared term is always positive (for \( a > 0 \)), the output is restricted above \( k \). Range is expressed as \( \{y | y > k\} \).
| # | Function | Key Features | Answer |
|---|
| 1 | \( f(x) = \frac{4}{x} \) | VA: \( x=0 \); HA: \( y=0 \) | B |
| 2 | \( g(x) = \frac{1}{x-3} + 2 \) | VA: \( x=3 \); HA: \( y=2 \) | C |
| 3 | \( h(x) = \frac{-5}{2x+6} - 1 \) | VA: \( x=-3 \); HA: \( y=-1 \) | A |
| 4 | \( f(x) = \frac{x-4}{x+5} \) | VA: \( x=-5 \); HA: \( y=1 \) | C |
| 5 | \( g(x) = \frac{2x+6}{x-1} \) | VA: \( x=1 \); HA: \( y=2 \) | A |
| 6 | \( h(x) = \frac{3x-9}{x+2} \) | VA: \( x=-2 \); HA: \( y=3 \) | B |
| 7 | \( f(x) = \frac{10}{(x-4)^2} + 5 \) | VA: \( x=4 \); Range: \( y > 5 \) | A |
| 8 | \( g(x) = \frac{-2}{x+8} - 4 \) | VA: \( x=-8 \); HA: \( y=-4 \) | B |
| 9 | \( h(x) = \frac{4x}{x-2} \) | VA: \( x=2 \); HA: \( y=4 \) | A |
| 10 | \( f(x) = \frac{1}{x+10} + 7 \) | VA: \( x=-10 \); HA: \( y=7 \) | D |
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