Domain, Asymptote, Undefined (Vertical)
CONCEPTS
Domain
• All the possible values of the independent variable, x, for which y is defined
Undefined values in rational functions
• Rational functions always take the form, y = p(x) / g(x), where p(x) & g(x) are both polynomials
• Undefined values in the domain are found by setting factors in the denominator = 0. Undefined values show up as asymptotes or discontinuities on a graph
• Vertical asymptotes: a line that continually approaches a given curve but does not meet it at any finite distance.
• Discontinuity: a break in a line that shows up as an open circle in a graph
• Undefined values in the domain are also found by determining the value under a square root is < 0
EXAMPLES
• y = (x – 3) / ((x – 3) (x – 4))
x = 3 & x = 4 are undefined values and they are not in the domain
x = 4 is a vertical asymptote ((x – 4) is a factor in only the denominator)
x = 3 is a discontinuity in the graph ((x – 3) is a factor in both the numerator & denominator)-it appears as an open circle in the graph
PRACTICE PROBLEMS
Question 1 x = {2, 3, 1, 3, 4, 6}. What is the domain of the set of numbers?
-- Answer -- 1, 2, 3, 4, 6
Question 2 Given function: y = 5x / ((x - 3)(x - 4))?
1-For what values of x is the function undefined?
2-What values of x are not in the domain in the function?
3-What is the linear equation that gives the vertical asymptote for the function?
(There are 3 ways to ask the question to get the same answer!)
-- Answer -- x = 3, x = 4
The factors are x - 3, x - 4
When the factors are set = 0, x - 3 = 0 -> x = 3 & x - 4 = 0 -> x = 4
The answers to 1, 2, 3 are all the same
Question 3 What is the linear equation that gives the vertical asymptote for (3x - 12) / (2x - 6)?
-- Answer -- 3
2x - 6 = 0, x = 3
Question 4 For what values of x is the function undefined: y = (x - 2) / (x² - 7x + 12)?
-- Answer -- x = 3, x = 4
The polynomial in factored form is y = (x - 2) / ((x - 3)(x - 4))
When the factors are set = 0, x - 3 = 0 -> x = 3 & x - 4 = 0 -> x = 4
Question 5 What's the vertical asymptote for the graph below?
-- Answer -- x = -2 & x = 2
Question 6 Given f(x) = (x - 2) / x² - 4 (see the graph below). What is the domain of x?
-- Answer -- The domain includes all real numbers except x = -2 and x = 2
Question 7 For what values of x is the function undefined: y = 5x / (x * √(x-4))?
-- Answer -- x = 0, x = 4, x < 4
Factors in the denominator: x, (x - 4) ¹ᐟ²
Undefined values: x = 0 & (x - 4) ¹ᐟ² = 0 -> x = 4, and (x - 4) ¹ᐟ² is a square root so x < 4 is also undefined
ACT PRACTICE PROBLEMS
