1. **State the problem:** Simplify and understand the function given by $y = e^{2x} \times \sqrt{4x^2 - 1}$. 2. **Recall the components:** - $e^{2x}$ is the exponential function with base $e$ raised to the power $2x$. - $\sqrt{4x^2 - 1}$ is the square root of the expression $4x^2 - 1$. 3. **Simplify inside the square root if possible:** The expression $4x^2 - 1$ is a difference of squares: $$4x^2 - 1 = (2x)^2 - 1^2 = (2x - 1)(2x + 1)$$ 4. **Rewrite the function:** $$y = e^{2x} \times \sqrt{(2x - 1)(2x + 1)}$$ 5. **Domain considerations:** The square root requires $4x^2 - 1 \geq 0$, so: $$4x^2 \geq 1 \implies x^2 \geq \frac{1}{4} \implies x \leq -\frac{1}{2} \text{ or } x \geq \frac{1}{2}$$ 6. **Final expression:** $$y = e^{2x} \sqrt{4x^2 - 1}$$ This is the simplified form, with domain restrictions as above. **Answer:** The function is $y = e^{2x} \sqrt{4x^2 - 1}$ with domain $x \leq -\frac{1}{2}$ or $x \geq \frac{1}{2}$.