The image you provided contains problems involving step functions (also called the "floor" function, denoted by ⌊x⌋, which gives the greatest integer less than or equal to $x$). Let’s analyze and solve each problem:

Problem 31:

$g(x) = -\lfloor x \rfloor$

• This function takes the floor of $x$ (the greatest integer less than or equal to $x$) and then negates it.
• Example values:
  • $g(1.5) = -\lfloor 1.5 \rfloor = -1$
  • $g(-0.7) = -\lfloor -0.7 \rfloor = 0$ (because $\lfloor -0.7 \rfloor = -1$, and negating gives 1)

The graph will look like a step function that steps downward as $x$ increases, with discontinuities at integer values.

Problem 32:

$g(x) = 4\lfloor x \rfloor$

• This function multiplies the floor of $x$ by 4.
• Example values:
  • $g(1.8) = 4\lfloor 1.8 \rfloor = 4 \times 1 = 4$
  • $g(-0.3) = 4\lfloor -0.3 \rfloor = 4 \times (-1) = -4$

This graph will also be step-like, but the height of each step will be 4 times the value of $\lfloor x \rfloor$.

Problem 33:

$g(x) = \lfloor x \rfloor - 1$

• This is the floor function minus 1.
• Example values:
  • $g(2.9) = \lfloor 2.9 \rfloor - 1 = 2 - 1 = 1$
  • $g(-1.2) = \lfloor -1.2 \rfloor - 1 = -2 - 1 = -3$

The graph will behave like the regular floor function but shifted downward by 1 unit.

Problem 34:

$g(x) = \lfloor x - 3 \rfloor$

• This is the floor of $x - 3$, which represents a horizontal shift of the floor function by 3 units to the right.
• Example values:
  • $g(4.5) = \lfloor 4.5 - 3 \rfloor = \lfloor 1.5 \rfloor = 1$
  • $g(2) = \lfloor 2 - 3 \rfloor = \lfloor -1 \rfloor = -1$

The graph of this function will be like the basic floor function but shifted to the right by 3 units.

Would you like detailed graph sketches for each of these functions, or further explanations?

Here are 5 relative questions to explore further:

1. How does adding or subtracting a constant outside the floor function affect the graph?
2. What happens when you multiply the floor function by a constant (as in problem 32)?
3. How would the graph of $g(x) = \lfloor x + 2 \rfloor$ differ from $g(x) = \lfloor x - 2 \rfloor$?
4. Can you describe how the discontinuities in step functions relate to their floor values?
5. How would the function $g(x) = -2\lfloor x \rfloor + 3$ behave?

Tip: Step functions always produce discontinuities at integer values of $x$. Understanding the behavior around these points is key to graphing them properly.