Algebra and functions

(a) Express (3√2 - 1)/(√2 + 1) in the form a + b√2, where a and b are integers. (b) Solve 8^x = 4^x+1.

Find the set of values of x for which x² - x - 6 > 0 and 2x + 1 < 9.

p(x) = 2x³ + x² - 13x + 6 (a) Use the factor theorem to show that (x - 2) is a factor of p(x). (b) Factorise p(x) completely. (c) Hence solve p(x) = 0 and sketch y = p(x), showing where it meets the axes.

The functions f and g are defined by f(x) = e^2x - 3 for x ∈ mathbb R, and g(x) = ln(x + 4) for x > -4. (a) State the range of f. (b) Find f^-1(x) and state its domain. (c) Find gf(x) in its simplest form. (d) Solve gf(x) = 2, giving your answer in exact form and to 3 decimal places. (e) Describe a sequence of two transformations that maps y = e^x onto y = f(x).