Practice sheet

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$f(x) = \dfrac{ 4x + 6 }{x - 6}$ has a vertical asymptote $x = c$ . What is $c$ ?
$f(x) = \dfrac{ 5x + 6 }{ 2x + 4 }$ has a horizontal asymptote $y = k$ . What is $k$ ?
What is the horizontal asymptote of $f(x) = \dfrac{ 4x + 1}{x^2 + 6}$ ?
1. $y = 0$
2. $y = 4$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 8x + 1}{x^2 + 5}$ ?
1. $y = 0$
2. $y = 8$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 7x + 1}{x^2 + 6}$ ?
1. $y = 0$
2. $y = 7$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 8x + 1}{x^2 + 6}$ ?
1. $y = 0$
2. $y = 8$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 8x + 1}{x^2 + 1}$ ?
1. $y = 0$
2. $y = 8$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 5x + 1}{x^2 + 4}$ ?
1. $y = 0$
2. $y = 5$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 6x + 1}{x^2 + 2}$ ?
1. $y = 0$
2. $y = 6$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 8x + 1}{x^2 + 3}$ ?
1. $y = 0$
2. $y = 8$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 3x + 1}{x^2 + 1}$ ?
1. $y = 0$
2. $y = 3$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 6x + 1}{x^2 + 6}$ ?
1. $y = 0$
2. $y = 6$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 5x + 1}{x^2 + 6}$ ?
1. $y = 0$
2. $y = 5$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 4x + 1}{x^2 + 4}$ ?
1. $y = 0$
2. $y = 4$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 9x + 1}{x^2 + 9}$ ?
1. $y = 0$
2. $y = 9$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 3x + 1}{x^2 + 6}$ ?
1. $y = 0$
2. $y = 3$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 6x + 1}{x^2 + 3}$ ?
1. $y = 0$
2. $y = 6$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 7x + 1}{x^2 + 9}$ ?
1. $y = 0$
2. $y = 7$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 3x + 1}{x^2 + 4}$ ?
1. $y = 0$
2. $y = 3$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 4x + 1}{x^2 + 2}$ ?
1. $y = 0$
2. $y = 4$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 9x + 1}{x^2 + 4}$ ?
1. $y = 0$
2. $y = 9$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 3x + 1}{x^2 + 8}$ ?
1. $y = 0$
2. $y = 3$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 7x + 1}{x^2 + 7}$ ?
1. $y = 0$
2. $y = 7$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 5x + 1}{x^2 + 9}$ ?
1. $y = 0$
2. $y = 5$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 2x + 1}{x^2 + 6}$ ?
1. $y = 0$
2. $y = 2$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 6x + 1}{x^2 + 8}$ ?
1. $y = 0$
2. $y = 6$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 9x + 1}{x^2 + 1}$ ?
1. $y = 0$
2. $y = 9$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 9x + 1}{x^2 + 6}$ ?
1. $y = 0$
2. $y = 9$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 2x + 1}{x^2 + 3}$ ?
1. $y = 0$
2. $y = 2$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 2x + 1}{x^2 + 7}$ ?
1. $y = 0$
2. $y = 2$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 4x + 1}{x^2 + 1}$ ?
1. $y = 0$
2. $y = 4$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 5x + 1}{x^2 + 1}$ ?
1. $y = 0$
2. $y = 5$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 3x + 1}{x^2 + 2}$ ?
1. $y = 0$
2. $y = 3$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 3x + 1}{x^2 + 3}$ ?
1. $y = 0$
2. $y = 3$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 2x + 1}{x^2 + 9}$ ?
1. $y = 0$
2. $y = 2$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 2x + 1}{x^2 + 1}$ ?
1. $y = 0$
2. $y = 2$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 9x + 1}{x^2 + 2}$ ?
1. $y = 0$
2. $y = 9$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 6x + 1}{x^2 + 7}$ ?
1. $y = 0$
2. $y = 6$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 7x + 1}{x^2 + 4}$ ?
1. $y = 0$
2. $y = 7$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 6x + 1}{x^2 + 4}$ ?
1. $y = 0$
2. $y = 6$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 5x + 1}{x^2 + 8}$ ?
1. $y = 0$
2. $y = 5$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 8x + 1}{x^2 + 7}$ ?
1. $y = 0$
2. $y = 8$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 7x + 1}{x^2 + 5}$ ?
1. $y = 0$
2. $y = 7$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 6x + 1}{x^2 + 1}$ ?
1. $y = 0$
2. $y = 6$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 8x + 1}{x^2 + 8}$ ?
1. $y = 0$
2. $y = 8$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 7x + 1}{x^2 + 8}$ ?
1. $y = 0$
2. $y = 7$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 2x + 1}{x^2 + 8}$ ?
1. $y = 0$
2. $y = 2$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 3x + 1}{x^2 + 5}$ ?
1. $y = 0$
2. $y = 3$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 9x + 1}{x^2 + 7}$ ?
1. $y = 0$
2. $y = 9$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 9x + 1}{x^2 + 3}$ ?
1. $y = 0$
2. $y = 9$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 9x + 1}{x^2 + 5}$ ?
1. $y = 0$
2. $y = 9$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 7x + 1}{x^2 + 2}$ ?
1. $y = 0$
2. $y = 7$
3. There is none, because the degrees are different
Where does the graph of $f(x) = \dfrac{x - 8}{x + 4}$ cross the $x$ -axis? Give $x$ .
$f(x) = \dfrac{(x - 5)(x + 6)}{x - 5}$ . At which $x$ does the graph have a hole?
$f(x) = \dfrac{(x - 1)(x + 3)}{x - 1}$ has a hole at $x = 1$ . What is the $y$ -coordinate of the hole?
Simplify (x² − 4) ÷ (x − 2), that is, the fraction with numerator x² − 4 and denominator x − 2. Which answer is right?
1. x + 2, for x ≠ 2
2. x − 2
3. x + 2, for every x
Simplify (x² − 25) ÷ (x − 5), that is, the fraction with numerator x² − 25 and denominator x − 5. Which answer is right?
1. x + 5, for x ≠ 5
2. x − 5
3. x + 5, for every x
Simplify (x² − 9) ÷ (x − 3), that is, the fraction with numerator x² − 9 and denominator x − 3. Which answer is right?
1. x + 3, for x ≠ 3
2. x − 3
3. x + 3, for every x
Simplify (x² − 16) ÷ (x − 4), that is, the fraction with numerator x² − 16 and denominator x − 4. Which answer is right?
1. x + 4, for x ≠ 4
2. x − 4
3. x + 4, for every x
Simplify (x² − 36) ÷ (x − 6), that is, the fraction with numerator x² − 36 and denominator x − 6. Which answer is right?
1. x + 6, for x ≠ 6
2. x − 6
3. x + 6, for every x
Simplify (x² − 81) ÷ (x − 9), that is, the fraction with numerator x² − 81 and denominator x − 9. Which answer is right?
1. x + 9, for x ≠ 9
2. x − 9
3. x + 9, for every x
Simplify (x² − 64) ÷ (x − 8), that is, the fraction with numerator x² − 64 and denominator x − 8. Which answer is right?
1. x + 8, for x ≠ 8
2. x − 8
3. x + 8, for every x
Simplify (x² − 49) ÷ (x − 7), that is, the fraction with numerator x² − 49 and denominator x − 7. Which answer is right?
1. x + 7, for x ≠ 7
2. x − 7
3. x + 7, for every x
Multiply and simplify: (x + 8)/(x − 6) · (x − 6)/(x + 9).
Divide and simplify: x/(x + 3) ÷ x²/(x + 3).
Write 1/x + 1/(x + 8) as a single fraction.
Simplify 6/(x − 1) − 9/(x − 1).
What is the least common denominator of 1/(x² − 16) and 1/(x + 4)?
Solve $\dfrac{x}{x + 2} = \dfrac{ 2 }{ 7 }$ .
Solve $\dfrac{1}{x} + \dfrac{1}{ 11 } = \dfrac{1}{ 5 }$ .
Solve $\dfrac{x^2 - 4}{x - 2} = 4$ .
1. No solution: simplifying gives x = 2, but x = 2 makes the denominator 0
2. x = 2
3. Every x except 2
Solve $\dfrac{x^2 - 81}{x - 9} = 18$ .
1. No solution: simplifying gives x = 9, but x = 9 makes the denominator 0
2. x = 9
3. Every x except 9
Solve $\dfrac{x^2 - 9}{x - 3} = 6$ .
1. No solution: simplifying gives x = 3, but x = 3 makes the denominator 0
2. x = 3
3. Every x except 3
Solve $\dfrac{x^2 - 16}{x - 4} = 8$ .
1. No solution: simplifying gives x = 4, but x = 4 makes the denominator 0
2. x = 4
3. Every x except 4
Solve $\dfrac{x^2 - 49}{x - 7} = 14$ .
1. No solution: simplifying gives x = 7, but x = 7 makes the denominator 0
2. x = 7
3. Every x except 7
Solve $\dfrac{x^2 - 36}{x - 6} = 12$ .
1. No solution: simplifying gives x = 6, but x = 6 makes the denominator 0
2. x = 6
3. Every x except 6
Solve $\dfrac{x^2 - 64}{x - 8} = 16$ .
1. No solution: simplifying gives x = 8, but x = 8 makes the denominator 0
2. x = 8
3. Every x except 8
Solve $\dfrac{x^2 - 25}{x - 5} = 10$ .
1. No solution: simplifying gives x = 5, but x = 5 makes the denominator 0
2. x = 5
3. Every x except 5
Solve $\dfrac{x^2 - 1}{x - 1} = 2$ .
1. No solution: simplifying gives x = 1, but x = 1 makes the denominator 0
2. x = 1
3. Every x except 1
Solve $\sqrt{x + 2} = 8$ .
Solve $\sqrt{x + 6} = x$ . (Squaring gives two candidates; enter the one that works.)
Solve $\sqrt[3]{x - 8} = 3$ .

Answers

$6$
$\frac{5}{2}$
$y = 0$
$8$
$5$
$4$
x + 2, for x ≠ 2
$\frac{x + 8}{x + 9}$
$\frac{1}{x}$
$\frac{2 x + 8}{x \left(x + 8\right)}$
$- \frac{3}{x - 1}$
$x^{2} - 16$
$\frac{4}{5}$
$\frac{55}{6}$
No solution: simplifying gives x = 2, but x = 2 makes the denominator 0
$62$
$3$
$35$