f : x → ax + b, x ∈ R, x > 0 and a < 0

The function \( f : x \mapsto ax + b \) is a linear function, where \( x \in \mathbb{R} \) and \( x > 0 \). Additionally, it is given that \( a < 0 \), meaning the slope of the line is negative.

Let's break this down:

1. **Domain**: The domain is restricted to positive real numbers, i.e., \( x > 0 \).
   
2. **Slope**: Since \( a < 0 \), the function decreases as \( x \) increases. This implies that the function is strictly decreasing for \( x > 0 \).
   
3. **Intercept**: The term \( b \) represents the y-intercept. Since \( x > 0 \), the intercept \( b \) influences the vertical translation of the graph.

### Behavior of the function:
- As \( x \) increases, the value of \( ax + b \) decreases due to the negative slope \( a \). 
- Depending on the value of \( b \), the graph may cross the x-axis or stay above/below it.

Would you like more details or have any questions about this function's behavior?

Here are 5 related questions to think about:

1. What happens to the function when \( b = 0 \)?
2. How does changing the value of \( b \) affect the graph?
3. Can the function have any roots (intersections with the x-axis)?
4. What is the behavior of the function as \( x \to \infty \)?
5. How would the graph look for different negative values of \( a \)?

**Tip**: To better understand the function's behavior, plot it for different values of \( a \) and \( b \).

Explore the linear function f(x) = ax + b, where the slope is negative (a < 0) and the domain is restricted to positive real numbers (x > 0). Understand how the function behaves as x increases, including intercepts and graph behavior. Suitable for Grades 9-11.

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