Theory question 113
Algebraic Expressions(a) Two functions, f and g, are defined by f : x → 2x2 − 1 and g : x → 3x + 2 where x is a real number. (i) If f(x − 1) − 7 = 0, find the values of x. (ii) Evaluate: f(−½) . g(3)/(f(4) − g(5)).
(b) An operation (∗) is defined on the set R, of real numbers by m ∗ n = −n/(m2 + 1), where m, n ∈ R. If −3, −10 ∈ R, show whether or not (∗) is commutative.
Show the model answer
Model answer
(a)(i) 2(x − 1)2 − 1 − 7 = 0 → (x − 1)2 = 4 → x − 1 = ±2 → x = 3 or x = −1.
(ii) f(−½) = −½, g(3) = 11, f(4) = 31, g(5) = 17. (−½ × 11)/(31 − 17) = −11/28.
(b) −3 ∗ −10 = 10/(9 + 1) = 1. −10 ∗ −3 = 3/(100 + 1) = 3/101.
Since 1 ≠ 3/101, the operation is not commutative.
Explanation
An operation is commutative only if swapping the two numbers never changes the result.