Question

12. (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∘f(x).

Accepted Answer

(a) Using the difference of two squares and rationalising, the expression simplifies to 0 + (736/81)√7, i.e. p=0 and q=736/81.
(b)(i) For f(x) to be real, (16-x²)/(x²+4) ≥ 0. Since x²+4>0 always, we need 16-x²≥0, i.e. -4≤x≤4. Domain: {x : -4 ≤ x ≤ 4}.
(ii) g∘f(x) = g(f(x)) = [f(x)²-4]/[8-f(x)²]. Substituting f(x)²=(16-x²)/(x²+4) and simplifying gives g∘f(x) = -5x²/(9x²+16).

WAEC Further Mathematics 2025 Theory — Question 12

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