Worksheet for identifying vertical and horizontal asymptotes, domain, and range of rational functions, with graphing exercises.

Let’s solve each problem step by step. We’ll find:

- Vertical asymptote (VA): where the denominator = 0 (and numerator ≠ 0)
- Horizontal asymptote (HA): what y approaches as x gets very large or very small
- Domain: all x-values that are allowed (exclude VA)
- Range: all y-values the function can output (exclude HA if it’s never reached)

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Problem 1: f(x) = -4/x

Step 1: Vertical Asymptote
Denominator is x → set x = 0 → VA at x = 0

Step 2: Horizontal Asymptote
As x → ∞ or -∞, -4/x → 0 → HA at y = 0

Step 3: Domain
Cannot divide by zero → x ≠ 0 → Domain: all real numbers except 0

Step 4: Range
The function never equals 0 (since -4/x = 0 has no solution) → Range: all real numbers except 0

✔ Final Answer for #1:
Vertical Asymptote: x = 0
Horizontal Asymptote: y = 0
Domain: x ≠ 0
Range: y ≠ 0

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Problem 2: f(x) = 4/(x - 1) + 1

Step 1: Vertical Asymptote
Denominator: x - 1 = 0 → x = 1 → VA at x = 1

Step 2: Horizontal Asymptote
As x → ±∞, 4/(x-1) → 0 → so f(x) → 0 + 1 = 1 → HA at y = 1

Step 3: Domain
x ≠ 1 → Domain: all real numbers except 1

Step 4: Range
f(x) = 1 only if 4/(x-1) = 0 → impossible → so y ≠ 1 → Range: all real numbers except 1

✔ Final Answer for #2:
Vertical Asymptote: x = 1
Horizontal Asymptote: y = 1
Domain: x ≠ 1
Range: y ≠ 1

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Problem 3: f(x) = -3/(x - 1) - 1

Step 1: Vertical Asymptote
Denominator: x - 1 = 0 → x = 1 → VA at x = 1

Step 2: Horizontal Asymptote
As x → ±∞, -3/(x-1) → 0 → f(x) → 0 - 1 = -1 → HA at y = -1

Step 3: Domain
x ≠ 1 → Domain: all real numbers except 1

Step 4: Range
f(x) = -1 only if -3/(x-1) = 0 → impossible → so y ≠ -1 → Range: all real numbers except -1

✔ Final Answer for #3:
Vertical Asymptote: x = 1
Horizontal Asymptote: y = -1
Domain: x ≠ 1
Range: y ≠ -1

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Problem 4: f(x) = -3/x

Same structure as #1.

Step 1: VA → x = 0
Step 2: HA → y = 0 (as x→±∞, -3/x→0)
Step 3: Domain → x ≠ 0
Step 4: Range → y ≠ 0 (never reaches 0)

✔ Final Answer for #4:
Vertical Asymptote: x = 0
Horizontal Asymptote: y = 0
Domain: x ≠ 0
Range: y ≠ 0

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Problem 5: f(x) = 3/(x + 1) - 2

Step 1: VA → denominator x + 1 = 0 → x = -1 → VA at x = -1

Step 2: HA → as x→±∞, 3/(x+1)→0 → f(x)→0 - 2 = -2 → HA at y = -2

Step 3: Domain → x ≠ -1 → Domain: all real numbers except -1

Step 4: Range → f(x) = -2 only if 3/(x+1)=0 → impossible → y ≠ -2 → Range: all real numbers except -2

Sketching tip:
- Draw dashed lines at x = -1 and y = -2
- The graph will be in two pieces: one in top-right quadrant relative to asymptotes, one in bottom-left
- When x > -1, say x=0 → f(0)=3/1 -2 = 1 → point (0,1)
- When x < -1, say x=-2 → f(-2)=3/(-1)-2 = -3-2 = -5 → point (-2,-5)

✔ Final Answer for #5:
Vertical Asymptote: x = -1
Horizontal Asymptote: y = -2
Domain: x ≠ -1
Range: y ≠ -2

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Problem 6: f(x) = 3/(x + 1) + 2

Step 1: VA → x + 1 = 0 → x = -1 → VA at x = -1

Step 2: HA → as x→±∞, 3/(x+1)→0 → f(x)→0 + 2 = 2 → HA at y = 2

Step 3: Domain → x ≠ -1 → Domain: all real numbers except -1

Step 4: Range → f(x) = 2 only if 3/(x+1)=0 → impossible → y ≠ 2 → Range: all real numbers except 2

Sketching tip:
- Dashed lines at x = -1 and y = 2
- Try x=0 → f(0)=3/1 +2 = 5 → point (0,5)
- Try x=-2 → f(-2)=3/(-1)+2 = -3+2 = -1 → point (-2,-1)

✔ Final Answer for #6:
Vertical Asymptote: x = -1
Horizontal Asymptote: y = 2
Domain: x ≠ -1
Range: y ≠ 2

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Final Answer:

1) VA: x = 0, HA: y = 0, Domain: x ≠ 0, Range: y ≠ 0
2) VA: x = 1, HA: y = 1, Domain: x ≠ 1, Range: y ≠ 1
3) VA: x = 1, HA: y = -1, Domain: x ≠ 1, Range: y ≠ -1
4) VA: x = 0, HA: y = 0, Domain: x ≠ 0, Range: y ≠ 0
5) VA: x = -1, HA: y = -2, Domain: x ≠ -1, Range: y ≠ -2
6) VA: x = -1, HA: y = 2, Domain: x ≠ -1, Range: y ≠ 2