Functions
Properties of f(x)=|x|+{x}-[x]-1 and g(x)
MJAT_TS8_P1
Grade 12
Question:
Let $f(x)=|x|+\{x\}-[x]-1$ ($x\in\mathbb{R}$) and $g(x)=\frac{1}{2}\!\left(x+\frac{1}{x+1}+\frac{x-1}{x}\right)$ ($x\in\mathbb{R}\setminus\{0\}$). Which is/are true?
A) $f(x-1/2)$ is an even function
B) $f(x)-g(x)=0$ has 2 solutions
C) $f(x)$ is non-differentiable at 2 points in its domain
D) $f(x)-g(x)=0$ has 3 solutions
Step-by-Step Solution
Key Concept: Simplify $f(x)=|x|+\{x\}-[x]-1=2\max([x],\{x\})-1$ (since $|x|+\{x\}-[x]-1$ for $x\geq 0$: $x+\{x\}-[x]-1=2\{x\}-1+[x]-[x]+[x]=...$). Analyse non-differentiability and solutions of $f=g$.
B ✓, C ✓. Answer: B, C.
Correct Answer: BC