Let's solve the problem step-by-step based on the given function [tex]\( f \)[/tex]:
### (a) Finding the Domain of the Function
The domain of a function is the set of all possible values of [tex]\( x \)[/tex] for which the function is defined. For the given function:
[tex]\[
f(x) = \begin{cases}
2x & \text{if } x \neq 0 \\
3 & \text{if } x = 0
\end{cases}
\][/tex]
The function [tex]\( f \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. There are no restrictions on [tex]\( x \)[/tex] since the function provides values for every possible [tex]\( x \)[/tex], whether [tex]\( x = 0 \)[/tex] or [tex]\( x \neq 0 \)[/tex].
So, the domain of the function [tex]\( f \)[/tex] is:
[tex]\[
\boxed{(-\infty, \infty)}
\][/tex]
### (b) Locating the Intercepts
Y-Intercept:
The y-intercept is the point where [tex]\( x = 0 \)[/tex]. Substituting [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
f(0) = 3
\][/tex]
Thus, the y-intercept is:
[tex]\[
\boxed{3}
\][/tex]
X-Intercept:
The x-intercept is the point where [tex]\( f(x) = 0 \)[/tex]. For [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
\begin{cases}
2x = 0 & \text{if } x \neq 0 \\
3 = 0 & \text{if } x = 0
\end{cases}
\][/tex]
For [tex]\( x \neq 0 \)[/tex], solving [tex]\( 2x = 0 \)[/tex] gives [tex]\( x = 0 \)[/tex], but at [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 3 \neq 0 \)[/tex]. Therefore, there is no valid [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex].
Thus, there are no x-intercepts:
[tex]\[
\boxed{\text{None}}
\][/tex]
### (c) Graphing the Function
To graph the function, we consider the two cases:
1. When [tex]\( x \neq 0 \)[/tex], the graph of [tex]\( f(x) = 2x \)[/tex] is a straight line passing through the origin with a slope of 2.
2. At [tex]\( x = 0 \)[/tex], the function takes the value [tex]\( f(0) = 3 \)[/tex]. This is a point away from the line [tex]\( 2x \)[/tex].
Here's a rough sketch of the graph:
- For [tex]\( x \neq 0 \)[/tex], the line [tex]\( y = 2x \)[/tex] is drawn except at [tex]\( x = 0 \)[/tex].
- At [tex]\( x = 0 \)[/tex], there's an isolated point [tex]\((0, 3)\)[/tex].
### (d) Finding the Range
The range of a function is the set of all possible values of [tex]\( f(x) \)[/tex] that the function can take. From the function behavior:
- When [tex]\( x \neq 0 \)[/tex], [tex]\( f(x) = 2x \)[/tex] can take any value from [tex]\( -\infty \)[/tex] to [tex]\( \infty \)[/tex] except at 0.
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 3 \)[/tex].
Thus, the range of the function [tex]\( f \)[/tex] is all real numbers except at the value 0:
[tex]\[
\boxed{(-\infty, 0) \cup (0, \infty)}
\][/tex]
This completes the thorough step-by-step solution to the given problem.
