The function f is defined as follows.
f left parenthesis x right parenthesis equalsleft brace Start 2 By 1 Matrix 1st Row 1st Column 4 x 2nd Row 1st Column 2 EndMatrix Start 2 By 1 Matrix 1st Row 1st Column if x not equals 0 2nd Row 1st Column if x equals 0 EndMatrix
 
 
​(a) Find the domain of the function.
​(b) Locate any intercepts.
​(c) Graph the function.
​(d) Based on the​ graph, find the range.

The function \( f(x) \) is defined as follows:

\[
f(x) = \begin{cases} 
4x & \text{if } x \neq 0 \\
2 & \text{if } x = 0 
\end{cases}
\]

### (a) Domain of the function:
The domain of a function is the set of all possible values of \( x \) for which the function is defined. In this case, the function is defined for all values of \( x \), since it has a rule for \( x = 0 \) and another for \( x \neq 0 \).

Thus, the domain of \( f(x) \) is:
\[
\text{Domain} = (-\infty, \infty)
\]

### (b) Intercepts:
- **x-intercept:** To find the x-intercept, we set \( f(x) = 0 \) and solve for \( x \). 
\[
4x = 0 \quad \text{(for \( x \neq 0 \))}
\]
This gives \( x = 0 \). However, the value of \( f(0) = 2 \), so there is no x-intercept since \( f(x) \neq 0 \) when \( x = 0 \).

- **y-intercept:** The y-intercept occurs when \( x = 0 \). 
\[
f(0) = 2
\]
So the y-intercept is \( (0, 2) \).

### (c) Graph of the function:
- For \( x \neq 0 \), the function behaves like a linear function with slope 4, i.e., \( f(x) = 4x \).
- At \( x = 0 \), the function value is \( f(0) = 2 \).

So, the graph will have a line with slope 4 for all \( x \neq 0 \), but at \( x = 0 \), there will be a jump where the function takes the value 2. The line for \( f(x) = 4x \) passes through \( (0, 0) \), but there is a discontinuity at \( x = 0 \), where the point is \( (0, 2) \) instead.

### (d) Range of the function:
Based on the graph:
- For \( x \neq 0 \), the function \( f(x) = 4x \) can take all real values except \( f(0) = 2 \).
- At \( x = 0 \), \( f(x) = 2 \).

Thus, the range of the function is:
\[
\text{Range} = (-\infty, \infty)
\]

Would you like to see the graph or have any further details? 

---

Here are 5 related questions to deepen your understanding:
1. What would change in the graph if \( f(0) \) were a different value, say 5?
2. How would the domain be affected if the function included a term that limited the values of \( x \)?
3. What are the conditions for a function to be discontinuous, as is the case here at \( x = 0 \)?
4. What would be the range if the function were \( f(x) = 4x^2 \) instead of \( f(x) = 4x \)?
5. How would you describe the end behavior of this function?

**Tip:** Discontinuous functions like this one are common in piecewise functions, where different rules apply in different intervals. Always check the conditions carefully.

The function f is defined as follows. f(x) = {4x if x ≠ 0, 2 if x = 0}. (a) Find the domain of the function. (b) Locate any intercepts. (c) Graph the function. (d) Based on the graph, find the range.

Explore how to find the domain, locate intercepts, graph, and determine the range of the piecewise function f(x) = {4x if x ≠ 0, 2 if x = 0}. This problem covers key concepts in graphing, function analysis, and algebra, suitable for students in Grades 9-12.

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