Question

The range of $k$ for which $|x^2-1|=k$ has exactly two solutions is

• A. $k\in \left\{0\right\}\cup (1,\infty )$
• B. $k\in \left\{0\right\}$
• C. $k\in (1,\infty )$
• D. $k\in (0,\infty )$

Answer 1

The correct option is A $k\in \left\{0\right\}\cup (1,\infty )$

The graph of $y=x^2$ is


Shifting the graph of $y=x^2$ vertically downward by $1$, we get the graph of $y=x^2-1$


Taking mod, we get the graph $y=|x^2-1|$


So, $k=|x^2-1|$ has exactly two solutions when $k\in \left\{0\right\}\cup (1,\infty )$