1. **State the problem:** Determine if the function $x + (\sqrt{2}x)^2 + y - 2x^2 = 11$ is linear or nonlinear. 2. **Recall the definition of linear functions:** A function is linear if it can be written in the form $ax + by = c$ where $a$, $b$, and $c$ are constants, and the variables appear only to the first power (no exponents other than 1) and are not multiplied together. 3. **Simplify the given function:** $$x + (\sqrt{2}x)^2 + y - 2x^2 = 11$$ Calculate $(\sqrt{2}x)^2$: $$(\sqrt{2}x)^2 = (\sqrt{2})^2 \cdot x^2 = 2x^2$$ Substitute back: $$x + 2x^2 + y - 2x^2 = 11$$ Simplify terms: $$x + y + (2x^2 - 2x^2) = 11$$ $$x + y = 11$$ 4. **Analyze the simplified equation:** The equation reduces to $x + y = 11$, which is linear because variables $x$ and $y$ are to the first power and not multiplied by each other. 5. **Conclusion:** The original function simplifies to a linear equation, so it represents a linear function.