Algebra and Functions
Functions and algebra
The Algebra and Functions domain tests whether you can manipulate functions, quadratic and polynomial expressions, indices, surds, logarithms, and sequences with confidence. Questions often combine two of these ideas, so the algebra must be fluent before you attempt the context.
Core rules
- For f(x), the domain is the set of allowed inputs and the range is the set of outputs. f-inverse exists only when f is one-to-one; to find it, swap x and y and solve for y.
- (f composed with g)(x) = f(g(x)): apply the inner function first.
- Completing the square writes ax^2 + bx + c as a(x - h)^2 + k, revealing the vertex (h, k). The discriminant b^2 - 4ac decides the number of real roots: positive, zero, negative gives two, one, none.
- Remainder theorem: the remainder when P(x) is divided by x - a is P(a). Factor theorem: x - a is a factor exactly when P(a) = 0.
- Log laws: log_a(xy) = log_a x + log_a y, log_a(x/y) = log_a x - log_a y, log_a x^n = n log_a x. Change of base: log_a b = log_c b / log_c a.
- AP: T_n = a + (n - 1)d, S_n = n/2[2a + (n - 1)d]. GP: T_n = ar^(n-1), S_n = a(r^n - 1)/(r - 1), and S_infinity = a/(1 - r) only for |r| < 1.
Worked example
Solve 3^(2x) - 10(3^x) + 9 = 0.
Let y = 3^x. Then 3^(2x) = (3^x)^2 = y^2, so the equation becomes y^2 - 10y + 9 = 0. Factor: (y - 1)(y - 9) = 0, so y = 1 or y = 9. Since 3^0 = 1 and 3^2 = 9, x = 0 or x = 2. Both check in the original equation, because 3^x is always positive.
Traps
Do not take logarithms of a negative number or zero. When solving a quadratic in 3^x, reject any negative value of y. In an inverse function question, state the domain restriction if the original function is not one-to-one. For a GP, do not use S_infinity unless |r| < 1, and do not confuse T_n with S_n. In the factor theorem, substitute the value that makes the bracket zero - for x + 2, use P(-2), not P(2).