Week 5 · Common domain and equivalent rational algebraic functions · 3 lessons
1
The common domain of two or more rational functions: R − (Z₁ ∪ Z₂)
Mathematics Prep 3 Term 1, Unit 2, Lesson 4
GoalFind the common domain of two or three rational functions from the zeros of their denominators.
New words
common domain · the x-values at which every one of the functions is defined: Df ∩ Dgintersection ∩ · the elements that belong to both setsunion ∪ · all the elements that belong to at least one of the setszeros of the denominator · the values that make the denominator 0; they are removed from the domain
Explanation
Sometimes we use two functions together, to add them, compare them or divide them. We then need the values of x where both are defined: the common domain. Write each denominator, find its zeros, call them Z₁ for f and Z₂ for g, and remove all of them: Df ∩ Dg = R − (Z₁ ∪ Z₂). Why a union of zeros? A number is lost if it is a bad value for f or for g (or for both), so we remove all the bad values together. It is the intersection of the domains, not their union. For three functions, take the union of three sets of zeros. A denominator with no real zero, like x² + 4 or x² + 2x + 4, removes nothing.
Function
Denominator
Zeros
Domain
f(x) = 1 ÷ (x − 2)
x − 2
Z₁ = {2}
R − {2}
g(x) = x ÷ (x² − 9)
x² − 9
Z₂ = {3, −3}
R − {3, −3}
common domain
Z₁ ∪ Z₂ = {−3, 2, 3}
R − {−3, 2, 3}
Worked examples
Find the common domain of f(x) = 1 ÷ (x − 2) and g(x) = x ÷ (x² − 9). Z₁ = {2}. For g: x² − 9 = (x − 3)(x + 3), so Z₂ = {3, −3}. Z₁ ∪ Z₂ = {−3, 2, 3}, so the common domain is R − {−3, 2, 3}. The picture shows the number line with these three values removed.
Find the common domain of f(x) = (x + 1) ÷ (x² − x) and g(x) = 3 ÷ (x − 1). For f: x² − x = x(x − 1), so Z₁ = {0, 1}. For g: Z₂ = {1}. Z₁ ∪ Z₂ = {0, 1}, so the common domain is R − {0, 1}. The value 1 is a bad value for both; we write it once.
Find the common domain of f(x) = 5 ÷ (x² + 4), g(x) = 1 ÷ (x + 7) and h(x) = x ÷ (x − 1). x² + 4 is never 0, so Z₁ = ∅. Z₂ = {−7} and Z₃ = {1}. The union is {−7, 1}, so the common domain is R − {−7, 1}.
Class activity
Number-line game. Draw a number line from −5 to 5. For f(x) = 1 ÷ (x − 1) put a ring at 1; for g(x) = 1 ÷ (x + 4) put a ring at −4. Colour everything else. Write the common domain, then add a third function of your own.
Practice
1
Find the common domain of 1 ÷ (x − 1) and 2 ÷ (x + 4).
Zeros {1} and {−4}. Common domain R − {1, −4}.
2
Find the common domain of x ÷ (x² − 25) and 1 ÷ (x² + 9).
Zeros {5, −5} and ∅. Common domain R − {5, −5}.
3
Find the common domain of 3 ÷ (x² − x) and x ÷ (x² − 4x + 3).
x² − x = x(x − 1): {0, 1}. x² − 4x + 3 = (x − 1)(x − 3): {1, 3}. Union {0, 1, 3}. Common domain R − {0, 1, 3}.
4
Find the common domain of 1 ÷ x, 1 ÷ (x − 1) and 1 ÷ (x − 2).
Zeros {0}, {1}, {2}. Common domain R − {0, 1, 2}.
5
Is 1 in the common domain of 1 ÷ (x − 1) and 1 ÷ (x − 2)? Explain.
No. The first function is undefined at 1, so both cannot be defined there. The common domain is R − {1, 2}.
