To find the $x$-intercepts of the function $g(x) = -x^2 + 4x - 3$, we need to solve the equation:

$g(x) = 0 \implies -x^2 + 4x - 3 = 0$

Rewriting the equation in standard form:

$x^2 - 4x + 3 = 0$

We can factor this quadratic equation:

$(x - 1)(x - 3) = 0$

Setting each factor equal to zero gives:

$x - 1 = 0 \implies x = 1$ $x - 3 = 0 \implies x = 3$

Therefore, the function has two $x$-intercepts at $x = 1$ and $x = 3$. The ordered pairs for these intercepts are $(1, 0)$ and $(3, 0)$.

The function has two $x$-intercepts.

Would you like further details or have any other questions?

Here are five related questions to deepen understanding:

1. How would the $x$-intercepts change if the function was $g(x) = -x^2 + 4x + 1$?
2. What are the $x$-intercepts of the function $g(x) = x^2 - 4x + 4$?
3. How can the quadratic formula be used to find the $x$-intercepts if the function cannot be easily factored?
4. What do the coefficients in a quadratic function tell us about the shape and position of its graph?
5. How do you determine the vertex of the parabola represented by $g(x) = -x^2 + 4x - 3$?

Tip: When factoring quadratics, always check if the equation can be simplified first, making the factoring process easier.