Functions
Discontinuities and root sum of composite functions
MJAT_TS2_P2
Grade 12
Question:
Let $\alpha$ denote the number of points of discontinuity of $f(x) = \dfrac{x+3}{x-1} + \dfrac{x+3}{|x|-1}$, and let $\beta$ denote the sum of all roots of $g(x)-h(x)=0$ in $[-2,2]$, where $g(x)=2-|x-2|$ and $h(x)=\dfrac{\sin x}{|\sin x|}$ ($x\neq 0$). Find $\alpha+\beta$.
Step-by-Step Solution
Key Concept: $f(x)$ has discontinuities at $x=1$ (from first term) and $x=\pm 1$ (from second term). So discontinuities at $x=-1$ and $x=1$: $\alpha=2$. For $g(x)=h(x)$: $h(x)=\text{sgn}(\sin x)=\pm 1$. $g(x)=2-|x-2|$. In $[-2,2]$: $g(x)=x$ (for $x\leq 2$). Solve $x=1$ (where $h=1$) and $x=-1$ (where $h=-1$). So $\beta=1+(-2)=-1$ or $\beta=1-2=-1$.
Discontinuities: $x=-1,1$, so $\alpha=2$. Roots: $x=1$ and $x=-1$, $\beta=0$. $\alpha+\beta=\mathbf{1}$.
Correct Answer: 1