Functions
Functions
Allen Star Batch
Grade 12
Question:
Define functions $f, g : \mathbb{R} \to \mathbb{R}$, $f(x) = 3x - 1 + |2x+1|$ and $g(x) = \frac{1}{5}(3x + 5 - |2x+5|)$ then
$f(g(x)) = g(f(x))$
$f(f(x)) = g(g(x))$
$f(g(x)) = x$
for $x < -\frac{5}{2}$, $f(g(x)) = x$
Step-by-Step Solution
Key Concept: Piecewise simplification of absolute value expressions by critical points: f(x) = 5x for x ≥ -1/2 and f(x) = x - 1 for x < -1/2; g(x) = x/5 for x ≥ -5/2 and g(x) = -x - 1 for x < -5/2. Function composition requires careful domain tracking across these pieces.
Given $f(x) = 5x$ for $x \geq -\frac{1}{5}$ and $g(x) = \frac{x}{5}$ for $x \geq -\frac{5}{2}$, we find compositions by considering domain restrictions. For $x 1$, and $f'(x) = \frac{2-x}{x^3}$ for $x > 1$.
Correct Answer: 1,3,4