Let's break down the problem into two parts: Part A involves finding specific characteristics of the function $f(x) = x - 3$, and Part B involves sketching the graph of the function.

Part A: Characteristics of the Function

1. Domain:

   • The function $f(x) = x - 3$ is a linear function, which means it is defined for all real numbers.
   • Domain: $(-\infty, \infty)$

2. Range:

   • Since the function is linear, it can produce any real number as an output depending on the input value.
   • Range: $(-\infty, \infty)$

3. Vertical Asymptote:

   • Vertical asymptotes occur where the function is undefined. However, linear functions are defined everywhere, so there are no vertical asymptotes.
   • Vertical Asymptote: None

4. Horizontal Asymptote:

   • Horizontal asymptotes are determined by the end behavior of the function. For linear functions like this one, there are no horizontal asymptotes since the function keeps increasing or decreasing without leveling off.
   • Horizontal Asymptote: None

5. x-intercept (x, 0):

   • To find the x-intercept, set $f(x) = 0$ and solve for $x$. $x - 3 = 0$ $x = 3$
   • x-intercept: $(3, 0)$

6. y-intercept (0, y):

   • To find the y-intercept, evaluate the function at $x = 0$. $f(0) = 0 - 3 = -3$
   • y-intercept: $(0, -3)$

7. Table of Values: Let's construct a table of values for $f(x)$ using several values of $x$:

   $x$	$f(x) = x - 3$
   -2	-5
   -1	-4
   0	-3
   1	-2
   2	-1
   3	0
   4	1
   5	2

Part B: Sketch the Graph

To sketch the graph of the function:

1. Plot the intercepts:
   • x-intercept at $(3, 0)$
   • y-intercept at $(0, -3)$
2. Draw the line:
   • Since $f(x) = x - 3$ is a straight line with a slope of 1 and y-intercept $-3$, draw a line passing through the points identified above.

I can generate a sketch of the graph for you if you'd like. Let me know!

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Would you like any details or have any questions? Here are five related questions that can help expand your understanding:

1. What is the general form of a linear function, and how can you identify the slope and y-intercept?
2. How do the domain and range change for a quadratic function compared to a linear function?
3. What happens to the intercepts if the function was $f(x) = x + 3$ instead of $f(x) = x - 3$?
4. Can a linear function have a horizontal or vertical asymptote? Why or why not?
5. How does the slope of the line affect the steepness of the graph?

Tip: The slope of a linear function determines whether the function is increasing or decreasing as you move from left to right on the graph. A positive slope means the function is increasing, while a negative slope means it's decreasing.