Let's consider the function [tex]\( f(x) = b^x \)[/tex] where [tex]\( 0 < b < 1 \)[/tex]. We will analyze each statement and determine its correctness.
1. The domain is all real numbers:
- Exponential functions of the form [tex]\( b^x \)[/tex] have no restrictions on [tex]\( x \)[/tex]. Therefore, the function is defined for all real numbers [tex]\( x \)[/tex].
- True
2. The domain is [tex]\( x > 0 \)[/tex]:
- As previously stated, [tex]\( b^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. There is no requirement that [tex]\( x \)[/tex] must be greater than 0.
- False
3. The range is all real numbers:
- For exponential functions where [tex]\( 0 < b < 1 \)[/tex], [tex]\( b^x \)[/tex] always yields positive results as [tex]\( b^x > 0 \)[/tex] for all real [tex]\( x \)[/tex]. Thus, the range does not cover all real numbers but only positive real numbers.
- False
4. The range is [tex]\( y > 0 \)[/tex]:
- As noted, when [tex]\( 0 < b < 1 \)[/tex], the output [tex]\( b^x \)[/tex] is always positive, meaning the range of [tex]\( f(x) \)[/tex] is [tex]\( y > 0 \)[/tex].
- True
5. The graph has [tex]\( x \)[/tex]-intercept 1:
- An [tex]\( x \)[/tex]-intercept occurs where the function crosses the x-axis, i.e., [tex]\( f(x) = 0 \)[/tex]. However, exponential functions [tex]\( b^x \)[/tex] (with [tex]\( 0 < b < 1 \)[/tex]) never equal zero, thus there is no [tex]\( x \)[/tex]-intercept.
- False
6. The graph has a [tex]\( y \)[/tex]-intercept of 1:
- A [tex]\( y \)[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex]. For [tex]\( f(x) = b^0 = 1 \)[/tex], the function always passes through (0, 1), so the [tex]\( y \)[/tex]-intercept is indeed 1.
- True
7. The function is always increasing:
- For a base [tex]\( 0 < b < 1 \)[/tex], [tex]\( f(x) = b^x \)[/tex] is a decreasing function because as [tex]\( x \)[/tex] increases, [tex]\( b^x \)[/tex] decreases.
- False
8. The function is always decreasing:
- Given [tex]\( 0 < b < 1 \)[/tex], as [tex]\( x \)[/tex] increases, [tex]\( b^x \)[/tex] decreases, making the function a decreasing function.
- True
Therefore, the true statements are:
- The domain is all real numbers.
- The range is [tex]\( y > 0 \)[/tex].
- The graph has a [tex]\( y \)[/tex]-intercept of 1.
- The function is always decreasing.
Thus, the answer is:
```
(True, False, False, True, False, True, False, True)
```
