We are asked to find a function $g(x)$ that represents two transformations applied to the graph of $f(x) = x + 1$:

1. A reflection in the x-axis.
2. A translation 3 units up.

Step-by-step solution:

1. Reflection in the x-axis:

   • Reflecting a graph in the x-axis means multiplying the output (or $y$-values) of the function by $-1$.
   • Applying this to $f(x) = x + 1$, we get: $g(x) = -(x + 1) = -x - 1$

2. Translation 3 units up:

   • To translate the graph 3 units upward, we add 3 to the function: $g(x) = (-x - 1) + 3$
   • Simplifying this: $g(x) = -x + 2$

Thus, the function after both transformations is $g(x) = -x + 2$.

Now, looking at the answer choices:

The correct option is g(x) = -x + 1 + 3, which simplifies to g(x) = -x + 2.

This corresponds to the third option in the list.

Do you have any questions or want further details?

Related Questions:

1. What would happen if the graph was translated down instead of up?
2. How would the function change if the reflection was in the y-axis instead of the x-axis?
3. What is the general form of a function reflection across the x-axis?
4. Can you describe how translations work with parabolas?
5. How do vertical stretches or compressions affect the graph of a function?

Tip: When performing reflections and translations, apply reflections first, then translations, as translations modify the position after the reflection transformation.