Functions f and g are defined by
\(f : x \mapsto 2x - 5, \quad x \in \mathbb{R},\)
\(g : x \mapsto \frac{4}{2-x}, \quad x \in \mathbb{R}, \; x \neq 2.\)
Find the value of \(x\) for which \(fg(x) = 7.\)
Solution
First, find \(fg(x)\) by applying \(g\) first, then \(f\):
\(g(x) = \frac{4}{2-x}\)
\(f(g(x)) = 2\left(\frac{4}{2-x}\right) - 5\)
Set \(f(g(x)) = 7\):
\(2\left(\frac{4}{2-x}\right) - 5 = 7\)
\(\frac{8}{2-x} - 5 = 7\)
\(\frac{8}{2-x} = 12\)
\(8 = 12(2-x)\)
\(8 = 24 - 12x\)
\(12x = 16\)
\(x = \frac{16}{12} = \frac{4}{3}\)
Log in to record attempts.