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Rational Functions and Their Properties Quiz

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#1

Which of the following is a rational function?

y = (3x + 2)/(x - 1)
Explanation

Ratio of two polynomials

#2

What is the domain of the rational function f(x) = (2x - 1)/(x + 3)?

All real numbers except x = -3
Explanation

Denominator cannot be zero

#3

Which of the following functions is not a rational function?

y = sin(x)
Explanation

Trigonometric functions are not ratios of polynomials

#4

What is the slant asymptote of the rational function f(x) = (3x^2 + 2)/(x + 1)?

y = 3x
Explanation

Occurs when degree of numerator > degree of denominator by 1

#5

If a rational function has a degree of 2 in the numerator and 3 in the denominator, what can be said about its end behavior?

It approaches positive infinity on both ends
Explanation

Leading term comparison

#6

What is the domain of the rational function f(x) = (x^2 - 9)/(x - 3)?

All real numbers except x = -3
Explanation

Denominator cannot be zero

#7

If the degree of the numerator is greater than the degree of the denominator in a rational function, what can be concluded about the end behavior?

It approaches positive infinity on both ends
Explanation

Leading term comparison

#8

What is the oblique asymptote of the rational function f(x) = (4x^2 + 2)/(x - 1)?

y = 4x + 2
Explanation

Occurs when degree of numerator > degree of denominator by 1

#9

What is the limit of the rational function (2x^2 - 5x + 3)/(x^2 + 2x - 3) as x approaches 1?

2
Explanation

Substitute the limiting value into the expression

#10

For the rational function f(x) = (x^2 + 4)/(x - 2), what is the vertical asymptote?

x = 2
Explanation

Vertical asymptote occurs where the denominator is zero

#11

Which statement is true about the vertical asymptotes of a rational function?

They occur where the denominator is zero
Explanation

Denominator determines vertical asymptotes

#12

If a rational function has a horizontal asymptote at y = 2, what can be inferred?

The function approaches 2 as x approaches infinity
Explanation

Horizontal asymptote defines limit behavior

#13

What is the end behavior of the rational function f(x) = (3x^2 + 2)/(2x^2 - x + 1)?

Approaches positive infinity as x approaches infinity
Explanation

Leading term comparison

#14

For the rational function g(x) = (x^2 - 4)/(x - 2), what is the hole's x-coordinate?

2
Explanation

Factor out common terms in numerator and denominator

#15

If a rational function has a vertical asymptote at x = -5 and a hole at x = 3, what can be said about the function?

The function has a removable discontinuity
Explanation

Removable discontinuity when both occur

#16

What is the degree of the numerator in the rational function h(x) = (2x^3 + 4x^2 + 1)/(x^2 - 1)?

3
Explanation

Degree of highest power term in numerator

#17

Which statement is true regarding the graph of a rational function with a hole?

The hole is a point of discontinuity
Explanation

Hole represents a jump in the graph

#18

For the rational function g(x) = (2x^2 - 5)/(x + 1), what is the horizontal asymptote?

y = 2
Explanation

Leading term comparison

#19

What is the behavior of a rational function as it approaches a vertical asymptote?

It approaches infinity or negative infinity
Explanation

Unbounded behavior towards vertical asymptote

#20

For the rational function h(x) = (x^2 - 1)/(x + 1), what is the value of the hole?

1
Explanation

Common factor canceled in both numerator and denominator

#21

If a rational function has a vertical asymptote at x = -4 and a hole at x = 2, what can be said about the function?

The function has a removable discontinuity
Explanation

Removable discontinuity when both occur

#22

What is the degree of the denominator in the rational function g(x) = (3x^3 + 2)/(2x^2 - x + 1)?

2
Explanation

Degree of highest power term in denominator

#23

Which statement is true about the behavior of a rational function near a hole?

The function approaches a finite value at the hole
Explanation

Hole represents a point where the function is defined but not continuous

#24

If a rational function has a horizontal asymptote at y = -3, what can be inferred?

The function approaches -3 as x approaches infinity
Explanation

Limit behavior of the function

#25

What is the end behavior of the rational function g(x) = (4x^3 - 2)/(2x^2 + 1)?

Approaches positive infinity as x approaches infinity
Explanation

Leading term comparison

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