1. **State the problem:** We are given a piecewise function: $$f(x) = \begin{cases} x^2 - 4x + 3, & x \neq 0 \\ 3, & x = 0 \end{cases}$$ 2. **Goal:** Understand the function and evaluate or analyze it, especially at $x=0$. 3. **Analyze the quadratic part:** For $x \neq 0$, the function is $f(x) = x^2 - 4x + 3$. 4. **Factor the quadratic:** $$x^2 - 4x + 3 = (x - 1)(x - 3)$$ 5. **Evaluate the function at $x=0$ using the quadratic expression:** $$f(0) = 0^2 - 4 \times 0 + 3 = 3$$ 6. **Check the piecewise definition at $x=0$:** It is given that $f(0) = 3$, which matches the quadratic evaluation. 7. **Conclusion:** The function is continuous at $x=0$ since both pieces agree there. **Final answer:** The function is $$f(x) = \begin{cases} x^2 - 4x + 3, & x \neq 0 \\ 3, & x = 0 \end{cases}$$ with $f(0) = 3$ consistent with the quadratic expression.