f(x)= 

​
  
2 x+5 if −3≤x<0
−3 if x=0
−5 x if 0<x   

​Find domain
 


The function \( f(x) \) is defined piecewise:

\[
f(x) = \begin{cases} 
2x + 5 & \text{if} \, -3 \leq x < 0 \\
-3 & \text{if} \, x = 0 \\
-5x & \text{if} \, 0 < x 
\end{cases}
\]

### Domain:

The domain of a function refers to all the possible values of \( x \) for which the function is defined.

1. For the first case, \( 2x + 5 \), the condition is \( -3 \leq x < 0 \), which includes all values of \( x \) from -3 to just below 0.
2. For the second case, \( -3 \), the function is defined at \( x = 0 \).
3. For the third case, \( -5x \), the condition is \( 0 < x \), which includes all values of \( x \) greater than 0.

### Conclusion:

The domain is the union of all these intervals, which means:

\[
\boxed{[-3, \infty)}
\]

This means the function is defined for all \( x \) starting from \( -3 \) (inclusive) and extends to infinity.

Would you like more details or clarifications? Here are 5 related questions:

1. What is the range of \( f(x) \)?
2. How do you graph a piecewise function?
3. What are the critical points of this function?
4. How do you evaluate \( f(x) \) at specific points in the domain?
5. How does continuity affect the graph of piecewise functions?

**Tip:** When dealing with piecewise functions, always check the boundaries carefully to understand the domain and behavior of each piece.

Find the domain of the piecewise function: f(x) = { 2x + 5 if −3 ≤ x < 0, -3 if x = 0, -5x if 0 < x }

Discover how to find the domain of a piecewise function with multiple conditions. This example explores key concepts in functions and domain analysis. Suitable for students in Grades 9-12, the solution covers detailed steps to solve the problem, including how to handle different function intervals.

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