Theory question 115
Inequalities(a) Given that f(x) = 2x − 1 and g(x) = x2 + 1: (i) find f(1 + x); (ii) find the range of values of x for which f(x) < −3; (iii) simplify f(x) − g(x).
(b) Given that A(1, 3), B(−2, −1) and C(2, 3m) where m is a constant; find: (i) |BA|; (ii) the value of m if BC = (4, −5).
Show the model answer
Model answer
(a)(i) f(1 + x) = 2(1 + x) − 1 = 2x + 1.
(ii) 2x − 1 < −3, so 2x < −2 and x < −1.
(iii) f(x) − g(x) = 2x − 1 − x2 − 1 = −x2 + 2x − 2.
(b)(i) |BA| = √[(1 + 2)2 + (3 + 1)2] = √25 = 5 units.
(ii) BC = (2 − (−2), 3m − (−1)) = (4, 3m + 1) = (4, −5), so 3m + 1 = −5 and m = −2.
Explanation
A vector from B to C is (position of C) − (position of B).