Theory question 111
Surds(a) Express [(4 + √7)/(4 − √7)]2 − [(4 − √7)/(4 + √7)]2 in the form p + q√7, where p and q are real numbers
(b) Two functions f and g are defined on the set of real numbers by f: x → √((16 − x2)/(x2 + 4)) and g: x → (x2 − 4)/(8 − x2), x ≠ ±2√2. Find: (i) the domain of f; (ii) g o f(x).
Show the model answer
Model answer
(a) (4 + √7)/(4 − √7) = (4 + √7)²/9 = (23 + 8√7)/9; its reciprocal is (23 − 8√7)/9.
Difference of squares: (a + b)(a − b) = (46/9)(16√7/9) = 736√7/81, so p = 0, q = 736/81.
(b)(i) Need 16 − x² ≥ 0 (x² + 4 > 0 always): domain {x : −4 ≤ x ≤ 4}.
(ii) [f(x)]² = (16 − x²)/(x² + 4). g(f(x)) = ([f]² − 4)/(8 − [f]²) = (−5x²/(x² + 4)) ÷ ((9x² + 16)/(x² + 4)) = −5x²/(9x² + 16).
Explanation
Spot that the two fractions are reciprocals and use a² − b² = (a + b)(a − b).