The function f is defined as follows.
f left parenthesis x right parenthesis equalsleft brace Start 2 By 1 Matrix 1st Row 1st Column 2 x 2nd Row 1st Column 1 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.
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Part 1
​(a) The domain of the function f is 
  
enter your response here.
​(Type your answer in interval​ notation.)

Let's go through each part of the problem step-by-step.

### (a) Domain of the function
The domain of a function is the set of all possible input values (x-values) for which the function is defined. The function \( f(x) \) is defined as:
\[ 
f(x) = 
\begin{cases} 
2x & \text{if } x \neq 0 \\ 
1 & \text{if } x = 0 
\end{cases}
\]

Since both \( 2x \) and 1 are defined for all real numbers, the function \( f(x) \) is defined for all real numbers \( x \).

So, the domain of \( f(x) \) is:
\[ 
(-\infty, \infty)
\]

### (b) Intercepts
#### x-intercept
The x-intercept is where the function crosses the x-axis, which means \( f(x) = 0 \).
- For \( x \neq 0 \), \( 2x = 0 \) implies \( x = 0 \), but for \( x = 0 \), the function value is \( 1 \).
So, there is no x-intercept.

#### y-intercept
The y-intercept is where the function crosses the y-axis, which is at \( x = 0 \).
- For \( x = 0 \), \( f(0) = 1 \).
So, the y-intercept is at \( (0, 1) \).

### (c) Graph of the function
To graph the function \( f(x) \), we note the following:
- For \( x \neq 0 \), \( f(x) = 2x \). This is a line with a slope of 2 passing through the origin, but we need to exclude the point at \( x = 0 \).
- For \( x = 0 \), \( f(x) = 1 \). This is a point at \( (0, 1) \).

Below is the plot:

1. For \( x \neq 0 \), the line is \( y = 2x \).
2. At \( x = 0 \), the function jumps to \( y = 1 \).

### (d) Range of the function
The range of a function is the set of all possible output values (y-values) of the function.

- For \( x \neq 0 \), \( y = 2x \) can take any real value.
- At \( x = 0 \), \( y = 1 \).

Thus, the range of the function \( f(x) \) includes all real numbers except for 1 because the function value 1 is explicitly defined only for \( x = 0 \). Hence, the range of the function is all real numbers:
\[ 
(-\infty, \infty)
\]

### Summary
(a) The domain of \( f \) is \( (-\infty, \infty) \).

(b) The function has no x-intercept and a y-intercept at \( (0, 1) \).

(c) The graph consists of the line \( y = 2x \) for \( x \neq 0 \) and a point at \( (0, 1) \).

(d) The range of \( f \) is \( (-\infty, \infty) \).

Do you want more details or have any questions about these solutions? Here are 5 related questions you might want to explore:

1. How do you find the domain of a more complex piecewise function?
2. Can you determine the range of a piecewise function with more than two pieces?
3. How do you graph a piecewise function with different types of functions (e.g., polynomial and exponential)?
4. How do intercepts of a function affect its graph?
5. What is the importance of continuity in piecewise functions?

**Tip:** When dealing with piecewise functions, always pay close attention to the conditions for each piece to ensure accurate graphing and analysis.

Explore the analysis of the piecewise function f(x) = {2x if x ≠ 0, 1 if x = 0}. Learn how to find its domain, intercepts, graph, and range with a detailed solution. This guide is suitable for high school students and covers key mathematical concepts.

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