The functions f and g are defined by f : x ↦ 7x - 1, x ∈ ℝ g : x ↦ 4/(x - 2), x ≠ 2, x ∈ ℝ (a) Solve the equation fg(x) = x (b) Hence, or otherwise, find the largest value of a such that g(a) = f⁻¹(a) - Edexcel - A-Level Maths Pure - Question 3 - 2016 - Paper 3

To solve the equation $fg(x) = x$, we first need to express $fg(x)$:

1. Substitute $g(x)$ into $f(x)$: $g(x) = \frac{4}{x - 2}$ Thus, substituting this in: $f(g(x)) = f\left(\frac{4}{x - 2}\right) = 7\left(\frac{4}{x - 2}\right) - 1$

2. Simplifying this expression: $f(g(x)) = \frac{28}{x - 2} - 1$ $= \frac{28 - (x - 2)}{x - 2}$ $= \frac{30 - x}{x - 2}$

3. Set this equal to $x$: $\frac{30 - x}{x - 2} = x$

4. Cross-multiply to eliminate the fraction: $30 - x = x(x - 2)$ $30 - x = x^2 - 2x$

5. Rearranging gives: $x^2 - x - 30 = 0$

6. This is a quadratic equation, which we can solve using the factorization method or the quadratic formula:

   • Factoring gives: $(x - 6)(x + 5) = 0$
   • This yields solutions: $x = 6$ or $x = -5$.