Common domain and equal rational functions
Answers are for parents and teachers.
1
Find the common domain of 5 ÷ (x − 4) and x ÷ (x² − 9).
Zeros {4} and {3, −3}. Common domain R − {4, 3, −3}.
2
Find the common domain of 1 ÷ (x² − 1), 2 ÷ (x + 5) and 4 ÷ (x² + 9).
Zeros {1, −1}, {−5} and ∅. Common domain R − {1, −1, −5}.
3
Are f(x) = (x² − x) ÷ (x² − 1) and g(x) = x ÷ (x + 1) equal? On which set do they agree?
f(x) = x(x − 1) ÷ ((x − 1)(x + 1)) = x ÷ (x + 1), Df = R − {1, −1}. Dg = R − {−1}. Not equal, but they agree on R − {1, −1}. g(1) = 1/2 exists, f(1) does not.
4
Solve f(x) = g(x) for f(x) = (x + 1) ÷ (x − 5) and g(x) = (3x − 7) ÷ (x − 5).
Common domain R − {5}. x + 1 = 3x − 7 gives x = 4, in the domain. Check: 5 ÷ (−1) = 5 ÷ (−1) ✓. Solution set {4}.
5
Is f(x) = (x² + 1) ÷ (x² + 1) equal to g(x) = 1? Explain.
Yes. x² + 1 is never 0, so Df = R = Dg, and f(x) = 1 everywhere.