Rational Functions Worksheet with problems on domain, asymptotes, and simplification.

Let’s solve each rational function problem step by step. For each, we’ll find:

a) x-intercepts → set numerator = 0 (and denominator ≠ 0)
b) vertical asymptotes → set denominator = 0 (and numerator ≠ 0)
c) horizontal or oblique asymptote → compare degrees of numerator and denominator

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

a) x-intercept: Set numerator = 0 → x + 3 = 0 → x = -3

b) Vertical asymptote: Set denominator = 0 → x - 5 = 0 → x = 5

c) Horizontal/oblique: Degree of num = 1, deg den = 1 → same degree → horizontal asymptote at y = leading coeff ratio = 1/1 → y = 1

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Problem 2: f(x) = 4x²/(x² - 1)

a) x-intercept: 4x² = 0 → x = 0 → x = 0

b) Vertical asymptote: x² - 1 = 0 → (x-1)(x+1)=0 → x = 1, x = -1

c) Degrees both 2 → horizontal asymptote: y = 4/1 → y = 4

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Problem 3: f(x) = (1 - x²)/(3x²) → rewrite as (-x² + 1)/(3x²)

a) x-intercept: -x² + 1 = 0 → x² = 1 → x = ±1 → x = 1, x = -1

b) Vertical asymptote: 3x² = 0 → x = 0 → x = 0

c) Degrees both 2 → horizontal asymptote: y = -1/3 → y = -1/3

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Problem 4: f(x) = (3x² + 4)/(x³ - 9x)

Factor denominator: x(x² - 9) = x(x - 3)(x + 3)

a) x-intercept: 3x² + 4 = 0 → x² = -4/3 → no real solution → no x-intercepts

b) Vertical asymptotes: x = 0, x = 3, x = -3 → x = 0, x = 3, x = -3

c) Degree num = 2, deg den = 3 → num < den → horizontal asymptote at y = 0

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Problem 5: f(x) = (4x² + x - 2)/(x² - 3x - 2)

Check if denominator factors? Discriminant: 9 + 8 = 17 → not factorable nicely → leave as is.

a) x-intercept: 4x² + x - 2 = 0 → use quadratic formula:

x = [-1 ± √(1 + 32)] / 8 = [-1 ± √33]/8 → x = (-1 + √33)/8, x = (-1 - √33)/8

b) Vertical asymptotes: x² - 3x - 2 = 0 → same roots: x = [3 ± √17]/2

Wait — let me recalculate discriminant for denominator: b² - 4ac = 9 - 4(1)(-2) = 9 + 8 = 17 → yes.

So vertical asymptotes: x = (3 + √17)/2, x = (3 - √17)/2

c) Degrees both 2 → horizontal asymptote: y = 4/1 → y = 4

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Problem 6: f(x) = (x² - 2)/(x² + x²)? Wait — look again.

Original says: f(x) = (x² - 2)/(x² + x²) → that would be (x² - 2)/(2x²)

But maybe it's a typo? Let me check the image description — user wrote: “f(x) = (x² - 2)/(x² + x²)” — but in standard problems, often it’s (x² - 2)/(x² + x) or similar.

Wait — looking back at original text: “6. f(x) = \frac{x^2 - 2}{x^2 + x^2}” — that simplifies to (x² - 2)/(2x²)

I’ll go with what’s written.

So f(x) = (x² - 2)/(2x²)

a) x-intercept: x² - 2 = 0 → x = ±√2 → x = √2, x = -√2

b) Vertical asymptote: 2x² = 0 → x = 0 → x = 0

c) Degrees both 2 → horizontal asymptote: y = 1/2 → y = 1/2

*(Note: If this was meant to be (x² - 2)/(x² + x), then denominator = x(x+1), VA at x=0,-1, etc. But based on given, we proceed.)*

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Problem 7: f(x) = (x - 1)/(1 - x²)

Note: 1 - x² = -(x² - 1) = -(x - 1)(x + 1)

So f(x) = (x - 1)/[-(x - 1)(x + 1)] = -1/(x + 1), provided x ≠ 1

So simplified: f(x) = -1/(x + 1), with hole at x = 1

a) x-intercept: After simplifying, numerator is -1 → never zero → no x-intercepts

But wait — before simplifying: original numerator x - 1 = 0 → x = 1, but at x=1, denominator also 0 → so it’s a hole, not intercept.

→ So no x-intercepts

b) Vertical asymptote: from simplified form, denominator x + 1 = 0 → x = -1 → x = -1

Also, original had denominator 1 - x² = 0 → x = ±1, but x=1 cancels → only VA at x=-1

c) Degree num = 1, deg den = 2 → num < den → horizontal asymptote y = 0

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Problem 8: f(x) = (x² - 4x - 12)/(x² + 4x - 60)

Factor both:

Numerator: x² - 4x - 12 = (x - 6)(x + 2)

Denominator: x² + 4x - 60 = (x + 10)(x - 6)

So f(x) = [(x - 6)(x + 2)] / [(x + 10)(x - 6)] = (x + 2)/(x + 10), for x ≠ 6

a) x-intercept: after simplifying, numerator x + 2 = 0 → x = -2 → x = -2

(Note: x=6 is a hole, not intercept)

b) Vertical asymptote: denominator x + 10 = 0 → x = -10 → x = -10

(canceled x-6, so no VA there)

c) Degrees both 1 → horizontal asymptote: y = 1/1 → y = 1

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Problem 9: f(x) = (-20x² - 45)/(8x² + 6x + 6)

First, factor out constants if possible.

Numerator: -5(4x² + 9) → doesn’t factor over reals

Denominator: 2(4x² + 3x + 3) → discriminant: 9 - 48 = -39 → no real roots

a) x-intercept: -20x² - 45 = 0 → 20x² = -45 → x² negative → no real x-intercepts

b) Vertical asymptotes: denominator 8x² + 6x + 6 = 0 → discriminant = 36 - 192 = -156 < 0 → no real roots → no vertical asymptotes

c) Degrees both 2 → horizontal asymptote: y = -20/8 = -5/2 → y = -2.5 or y = -5/2

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Now compile all answers clearly.

Final Answer:

1.
a) x = -3
b) x = 5
c) y = 1

2.
a) x = 0
b) x = 1, x = -1
c) y = 4

3.
a) x = 1, x = -1
b) x = 0
c) y = -1/3

4.
a) none
b) x = 0, x = 3, x = -3
c) y = 0

5.
a) x = (-1 ± √33)/8
b) x = (3 ± √17)/2
c) y = 4

6.
a) x = √2, x = -√2
b) x = 0
c) y = 1/2

7.
a) none
b) x = -1
c) y = 0

8.
a) x = -2
b) x = -10
c) y = 1

9.
a) none
b) none
c) y = -5/2