Functions f and g are defined by
\(f : x \mapsto 10 - 3x, \quad x \in \mathbb{R},\)
\(g : x \mapsto \frac{10}{3 - 2x}, \quad x \in \mathbb{R}, \; x \neq \frac{3}{2}.\)
Solve the equation \(ff(x) = gf(2)\).
Solution
First, calculate \(ff(x)\):
\(ff(x) = f(f(x)) = f(10 - 3x) = 10 - 3(10 - 3x)\)
\(= 10 - 30 + 9x = 9x - 20\).
Next, calculate \(gf(2)\):
\(f(2) = 10 - 3(2) = 10 - 6 = 4\).
\(gf(2) = g(f(2)) = g(4) = \frac{10}{3 - 2(4)} = \frac{10}{3 - 8} = \frac{10}{-5} = -2\).
Set \(ff(x) = gf(2)\):
\(9x - 20 = -2\).
Solve for \(x\):
\(9x = 18\)
\(x = 2\).
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