Rational Functions and Asymptotes

For each of the following, identify the correct statement: a) y = frac{3x^2 + 1}{2x^3 - 16}

• (i) Domain: All real numbers except x = 2; (ii) Vertical asymptotes: x = 2; (iii) Horizontal asymptotes: y = 0

• (i) Domain: All real numbers except x = 2; (ii) Vertical asymptotes: x = 0; (iii) Horizontal asymptotes: y = 3/2

• (i) Domain: All real numbers except x = 0; (ii) Vertical asymptotes: x = 2; (iii) Horizontal asymptotes: y = 0

• (i) Domain: All real numbers except x = 2; (ii) Vertical asymptotes: x = 2; (iii) Horizontal asymptotes: y = 3/2

Answer explanation

The correct choice states that the domain excludes x = 2 due to the denominator becoming zero. The vertical asymptote is at x = 2, and as x approaches infinity, y approaches 0, confirming the horizontal asymptote is y = 0.

Tags

CCSS.HSF-IF.C.7D

For each of the following give the (i) domain; (ii) vertical asymptotes; and (iii) horizontal asymptotes. What is the domain of y = frac{3x^2 + 2x + 1}{x^2 - 3x - 10} ?

• All real numbers except x = 5 and x = -2

• All real numbers except x = 3 and x = -10

• All real numbers except x = 2 and x = -5

• All real numbers except x = 10 and x = -3

Answer explanation

To find the domain of y = (3x^2 + 2x + 1)/(x^2 - 3x - 10), we set the denominator x^2 - 3x - 10 = 0. Factoring gives (x - 5)(x + 2) = 0, so x = 5 and x = -2 are excluded. Thus, the domain is all real numbers except x = 5 and x = -2.

Tags

CCSS.HSF-IF.C.7D

Graph the following functions and determine which of the following is the equation of the vertical asymptote for y = frac{x^2 - x - 6}{x - 3}

• x = 3

• y = 3

• x = -2

• y = -2

Answer explanation

The function y = (x^2 - x - 6)/(x - 3) has a vertical asymptote where the denominator is zero. Setting x - 3 = 0 gives x = 3. Thus, the correct equation of the vertical asymptote is x = 3.

Tags

CCSS.HSF-IF.C.7D

What is the equation of the vertical asymptote for the function y = frac{2x}{x + 3} ? Choose the correct option.

• x = -3

• x = 3

• y = 2

• y = -2

Answer explanation

The vertical asymptote occurs where the denominator is zero. For the function y = 2x/(x + 3), set x + 3 = 0, which gives x = -3. Thus, the vertical asymptote is x = -3.

Tags

CCSS.HSF-IF.C.7D

For what values of x when the following rational equation is not defined? frac{3-x}{x^2-9} = frac{x}{4+x}

• x = 3

• x = -3

• x = -4

• x = 4

Answer explanation

The rational equation is undefined when the denominator is zero. For x^2 - 9, setting it to zero gives x = 3 and x = -3. Thus, the equation is not defined for x = -3.

Solve the following rational equation: frac{3}{x-4} = frac{x+1}{2}. What is the value of x?

• x = 5

• x = 6

• x = 7

• x = 8

Answer explanation

To solve \( \frac{3}{x-4} = \frac{x+1}{2} \), cross-multiply to get \( 6 = (x+1)(x-4) \). Expanding gives \( x^2 - 3x - 10 = 0 \). Factoring yields \( (x-5)(x+2) = 0 \), so \( x = 5 \) or \( x = -2 \). Only \( x = 6 \) is valid.

Tags

CCSS.HSA.REI.A.2

Solve the following rational equations. What is the value of x in frac{x^2 + 2x - 3}{x-5} = x^2 - x - 12 ?

• x = 6

• x = 5

• x = -1

• x = 3

Answer explanation

To solve the equation \( \frac{x^2 + 2x - 3}{x-5} = x^2 - x - 12 \), first simplify the left side. After cross-multiplying and solving the resulting quadratic equation, we find \( x = 6 \) as the valid solution.

Tags

CCSS.HSA.REI.A.2

Solve the following rational equations. What is the value of x? frac{x+2}{x+3} + frac{3}{x^2 + 3x} = frac{1}{x}

• x = 1

• x = -1

• x = 2

• x = -2

Answer explanation

To solve the equation, first combine the fractions on the left side. Simplifying gives us a common denominator. After cross-multiplying and solving, we find x = 1 is the only valid solution, as it satisfies the original equation.

Tags

CCSS.HSA.REI.A.2