Question

Explain the process of finding the range of a continuous linear function with a domain of {x\a<x<b} where a and b are real numbers.

Answer 1

To find the range of a continuous linear function with a specified domain of the form \( \{ x \mid a < x < b \} \), we can follow these steps:

1. **Identify the Function**: A general linear function can be expressed in the form \( f(x) = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.

2. **Determine the Endpoints**: Since the domain is \( a < x < b \), we will evaluate the function at the endpoints \( a \) and \( b \). Even though \( a \) and \( b \) are not included in the domain, we will use their values to find the corresponding output.

3. **Evaluate the Function at the Endpoints**:
- Compute:
\[
f(a) = ma + c
\]
- Compute:
\[
f(b) = mb + c
\]

4. **Determine the Behavior of the Function**: Since the function is linear:
- If \( m > 0 \) (the slope is positive), the function will increase as \( x \) moves from \( a \) to \( b \), so \( f(a) < f(b) \).
- If \( m < 0 \) (the slope is negative), the function will decrease as \( x \) increases from \( a \) to \( b \), so \( f(a) > f(b) \).
- If \( m = 0 \) (the function is constant), then \( f(x) \) will be the same for all \( x \) in the domain, and the range will just be the single value \( f(a) = f(b) = c \).

5. **Define the Range**:
- For \( m > 0 \):
\[
\text{Range} = (f(a), f(b))
\]
- For \( m < 0 \):
\[
\text{Range} = (f(b), f(a))
\]
- For \( m = 0 \):
\[
\text{Range} = \{ c \}
\]

By following these steps, we can conclusively determine the range of a continuous linear function over the specified open interval \( (a, b) \).