A Value-Range Cascade

★ 8/10functions Quadratic Equations Substitution Casework

Problem

The function $g(x) = x^2 - 2ax + 1$ has range $[0, 4]$ on the interval $[1, 3]$ .

1. Find $a$ . 2. If $g\left(2^x\right) - k\cdot 4^x \geqslant 0$ for all $x \in [1, +\infty)$ , find the range of possible values of $k$ . 3. If the function

y = \frac{g\left(|2^x - 1|\right)}{|2^x - 1|} + k\cdot\frac{2}{|2^x - 1|} - 3k

has exactly $3$ zeros, find the range of possible values of $k$ .

Answer

Solution

Difficulty	8/10
Topics	functions, Quadratic Equations, Substitution, Casework

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