1. **State the problem:** We are given the function $$y = e^{\sqrt{7x^2 - 1}}$$ and want to understand its behavior. 2. **Recall the formula and rules:** The function involves an exponential with base $e$ raised to the power of the square root of an expression. Important rules: - The expression inside the square root, $7x^2 - 1$, must be non-negative for real values, so $$7x^2 - 1 \geq 0 \implies x^2 \geq \frac{1}{7} \implies x \leq -\frac{1}{\sqrt{7}} \text{ or } x \geq \frac{1}{\sqrt{7}}.$$ - The exponential function $e^u$ is always positive for any real $u$. 3. **Analyze the domain:** The domain of $y$ is $$(-\infty, -\frac{1}{\sqrt{7}}] \cup [\frac{1}{\sqrt{7}}, \infty).$$ 4. **Evaluate the function at domain boundaries:** - At $x = \pm \frac{1}{\sqrt{7}}$, inside the root is zero, so $$y = e^0 = 1.$$ 5. **Behavior for large $|x|$:** As $|x| \to \infty$, $7x^2$ dominates, so inside the root behaves like $\sqrt{7x^2} = \sqrt{7}|x|$, and thus $$y \approx e^{\sqrt{7}|x|},$$ which grows very rapidly. 6. **Summary:** The function is defined only outside the interval $$\left(-\frac{1}{\sqrt{7}}, \frac{1}{\sqrt{7}}\right)$$ and grows exponentially as $|x|$ increases beyond these points. Final answer: The domain is $$x \leq -\frac{1}{\sqrt{7}} \text{ or } x \geq \frac{1}{\sqrt{7}}$$ and $$y = e^{\sqrt{7x^2 - 1}}$$ grows exponentially for large $|x|$.