Consider the curves $f(x)=x|x|-1$ and $g(x)= \begin{cases}\frac{3 x}{2}, & x \gt 0 \\ 2 x, & x \leq 0\end{cases}$ Where do the curves intersect?

• A

  At $(2,3)$ only
• B

  At $(-1,-2)$ only
• Correct answer
  C

  $\operatorname{At}(2,3)$ and $(-1,-2)$
• D

  Neither at $(2,3)$ nor at $(-1,-2)$

Solution

$\begin{aligned} & f(x)=x|x|-1 \\ & f(x)=\left\{\begin{array}{ll} x^2-1 & x \gt 0 \\ -x^2-1 & x \leq 0 \end{array} \quad g(x)= \begin{cases}\frac{3 x}{2} & x \gt 0 \\ 2 x & x \leq 0\end{cases} \right. \end{aligned}$
Hence $f(x)$ and $g(x)$ intersects at $(-1,-2)$ and $(2,3)$.

Hence, the correct answer is option: 'C'.