Problem Statement
Let $f(x) = \sqrt{3x - 5}$ for $x \geq \frac{5}{3}$, and $g(x) = x^2 + 1$ for $x \in \mathbb{R}$.\n\n(a) Find an expression for $(f \circ g)(x)$. [2 marks]\n\n(b) Solve the equation $(f \circ g)(x) = 5$. [3 marks]
Verified Solution & Marking Scheme
Part (a): Evaluate composite function
$$(f \circ g)(x) = f(g(x)) = f(x^2 + 1) = \sqrt{3(x^2 + 1) - 5} = \sqrt{3x^2 + 3 - 5} = \sqrt{3x^2 - 2}$$
Part (b): Solve composite equation
$$\sqrt{3x^2 - 2} = 5 \implies 3x^2 - 2 = 25 \implies 3x^2 = 27 \implies x^2 = 9 \implies x = \pm 3$$
Check domain validity
For $$x = \pm 3$$, $$g(x) = (\pm 3)^2 + 1 = 10 \geq \frac{5}{3}$$. Both solutions are valid.