The function given is:

$f(x) = \left\lfloor \frac{3}{4}x \right\rfloor$

This represents a floor function, where $\left\lfloor \cdot \right\rfloor$ rounds down to the nearest integer.

Domain:

The domain of this function is all real numbers because the function is defined for any real value of $x$. Thus, the domain is:

$\text{Domain: } (-\infty, \infty)$

Range:

The range of the floor function is the set of all integers, because the floor function returns integer values for any real input. However, for this specific function, the values depend on the multiplication by $\frac{3}{4}$. The result is still all integers, but we can state that:

$\text{Range: } \mathbb{Z} \text{ (all integers)}$

Graph of $f(x) = \left\lfloor \frac{3}{4} x \right\rfloor$:

The graph of the floor function $\left\lfloor \frac{3}{4}x \right\rfloor$ is a step function. Each "step" occurs at points where $x$ makes $\frac{3}{4}x$ an integer. Between each step, the value of the function remains constant, and it decreases by 1 as $x$ increases to the next integer multiple of $\frac{4}{3}$.

Here is a rough outline of how the graph looks:

• The function steps up at points like $x = 0, \frac{4}{3}, \frac{8}{3}, \dots$
• Between these points, the value of the function is constant.

I will now graph the function for better visualization.Here is the graph of the function $f(x) = \left\lfloor \frac{3}{4}x \right\rfloor$. As you can see, the graph consists of steps that occur at intervals, where the value of the function remains constant between jumps.

Summary:

• Domain: $(-\infty, \infty)$
• Range: $\mathbb{Z}$ (all integers)

Do you have any questions or need further details?

Here are 5 related questions for deeper understanding:

1. How would the graph change if the function were $f(x) = \left\lceil \frac{3}{4}x \right\rceil$ (using the ceiling function)?
2. What is the effect of changing the coefficient $\frac{3}{4}$ to another number, say $\frac{2}{3}$, in the function?
3. How does the domain and range change for the function $f(x) = \left| \frac{3}{4}x \right|$?
4. Can you find where the function $f(x) = \left\lfloor \frac{3}{4}x \right\rfloor$ is constant?
5. What happens if we modify the function to $f(x) = -\left\lfloor \frac{3}{4}x \right\rfloor$?

Tip: The floor function always rounds down to the greatest integer less than or equal to the input value.