Rational algebraic fractions: when the denominator is zero, and the domain R − {zeros of the denominator}
Mathematics Prep 3 Term 1, Unit 2, Lesson 3
GoalRecognise a rational algebraic fraction, find where it is undefined, and write the domain of the function.
New words
rational algebraic fraction · p(x) ÷ q(x), where p and q are polynomials and q(x) ≠ 0rational function · f(x) = p(x) ÷ q(x); every polynomial function is one, with q(x) = 1domain Df· the set of x-values for which the function has a valueundefined · has no value, as with division by zero
Explanation
A rational algebraic fraction is a polynomial divided by a polynomial, such as (x + 1) ÷ (x² − 5x + 6). It is not rational if x is under a root or in an exponent: √x ÷ 2 and 3 ÷ 2ˣ do not count. Division by zero has no meaning, so the fraction is undefined at every x that makes the denominator 0. To find these values, solve denominator = 0. For x² − 5x + 6 = (x − 2)(x − 3) the values are 2 and 3. The function f(x) = p(x) ÷ q(x) therefore has the domain Df = R − {zeros of q}, all real numbers except those. If the denominator has no real zero, as in x² + 16, nothing is removed and Df = R. Every polynomial function is a rational function with denominator 1, and its domain is R.
Fraction
Denominator = 0
Domain
(x + 1) ÷ (x² − 5x + 6)
x = 2 or 3
R − {2, 3}
3x ÷ (x² − 9)
x = −3 or 3
R − {−3, 3}
7 ÷ (x² + 16)
none
R
x² + 1
denominator 1
R
Worked examples
Find the domain of f(x) = (x + 1) ÷ (x² − 5x + 6). Solve x² − 5x + 6 = 0: (x − 2)(x − 3) = 0, so x = 2 or x = 3. The domain is R − {2, 3}. The numerator x + 1 does not matter for the domain.
Find the domain of f(x) = 7 ÷ (x² + 16). x² + 16 is at least 16, so it is never 0. Nothing is removed and Df = R.
Let f(x) = (x² − 16) ÷ (x − 4). Find f(4), f(−4), f(0) and f(2), and the domain. The denominator is 0 at x = 4, so f(4) is undefined and Df = R − {4}. f(−4) = 0 ÷ (−8) = 0. f(0) = −16 ÷ (−4) = 4. f(2) = (4 − 16) ÷ (−2) = 6.
Class activity
Which is which? Write five fractions on slips: three rational algebraic fractions and two that are not (with a root or an x in an exponent). Swap with a partner and sort them. For each rational one, write the domain.
Practice
1
Is (x + 2) ÷ √x a rational algebraic fraction?
No. The denominator contains a root of x, so it is not a polynomial.
2
Find the domain of 5 ÷ (x − 7).
x − 7 = 0 at x = 7, so the domain is R − {7}.
3
Find the domain of x ÷ (x² − 4x).
x² − 4x = x(x − 4) = 0 at x = 0 and 4. The domain is R − {0, 4}.
4
Find the domain of 3 ÷ (x² + 1).
x² + 1 ≥ 1, never 0. The domain is R.
5
Let f(x) = (x² − 9) ÷ (x + 3). Find f(3), f(1) and f(−3).
f(3) = 0 ÷ 6 = 0. f(1) = (1 − 9) ÷ 4 = −2. f(−3) is undefined, since the denominator is 0.
