(a) Find the domain of each function. (b) Locate any interepts. (c) Graph each funct

step 1 We are given the function that f of x is equal to 2 multiplied with the greatest integer functio

(a) Find the domain of each function. (b) Locate...

$$ \begin{array}{ll}{\text { (a) Find the domain of each function. }} & {\text { (b) Locate any interepts. }}& {\text {(c) Graph each function.. }} \\ {\text { (d) Based on the graph, find the range. }} & {\text { (e) Is feontinuous on its domain? }}\end{array} $$ $$ f(x)=2 \operatorname{int}(x) $$

$$
\begin{array}{ll}{\text { (a) Find the domain of each function. }} & {\text { (b) Locate any interepts. }}& {\text {(c) Graph each function.. }} \\ {\text { (d) Based on the graph, find the range. }} & {\text { (e) Is feontinuous on its domain? }}\end{array}
$$
$$
f(x)=2 \operatorname{int}(x)
$$

Key Concepts

Continuity

Continuity in a function refers to the absence of any breaks, jumps, or holes in its graph. A continuous function has a graph that can be drawn without lifting the pen. However, many step functions or piecewise-defined functions inherently have discontinuities, particularly at the points where the function’s formula changes.

Graphing Piecewise or Step Functions

Graphing piecewise or step functions involves plotting distinct segments of the function over specific intervals. With a step function, the rule remains constant over intervals and then jumps to a different value at the boundaries. This requires careful notation of inclusion or exclusion at endpoints, typically using open or closed circles to indicate whether the endpoint is part of the graph.

Greatest Integer (or Floor) Function

The greatest integer function, often written as int(x) or ?x?, returns the largest integer less than or equal to x. This function is a classic example of a step function, as it remains constant within each interval from one integer to the next and then jumps to a new integer value. Its properties are essential for understanding behaviors such as discrete ranges and inherent discontinuities.

Range of a Function

The range of a function is the set of all possible outputs (or y-values) that the function can produce. For functions that have discrete outputs or are defined piecewise, the range may consist of a series of isolated numbers or intervals, which requires analyzing the function’s behavior across its entire domain.

Domain of a Function

The domain of a function represents all the input values (typically x-values) for which the function is defined. Determining the domain involves understanding the operations within the function and any restrictions (like division by zero or square roots of negative numbers), ensuring that every element in the domain results in a real number output.

Intercepts

Intercepts are the points where the graph of a function crosses the axes. The x-intercepts occur when the output (y-value) equals zero, indicating where the graph meets the x-axis, while the y-intercept is the value of the function when the input (x) is zero. Identifying intercepts is a key part of graphing any function.

Please give Ace some feedback