Answer :
Let's analyze the function [tex]\( f(x) = b^x \)[/tex] where [tex]\( 0 < b < 1 \)[/tex] and examine each statement one by one to determine their validity:
1. The domain is all real numbers:
- For the function [tex]\( f(x) = b^x \)[/tex] where [tex]\( 0 < b < 1 \)[/tex], [tex]\( x \)[/tex] can take any real value. There are no restrictions on [tex]\( x \)[/tex].
- Therefore, the statement "The domain is all real numbers" is True.
2. The domain is [tex]\( x > 0 \)[/tex]:
- As determined above, the domain of [tex]\( f(x) = b^x \)[/tex] includes all real numbers, not just [tex]\( x > 0 \)[/tex].
- Therefore, the statement "The domain is [tex]\( x > 0 \)[/tex]" is False.
3. The range is all real numbers:
- For [tex]\( f(x) = b^x \)[/tex] where [tex]\( 0 < b < 1 \)[/tex], the output (range) of [tex]\( f(x) \)[/tex] is always positive and never zero or negative.
- Therefore, the range is not all real numbers.
- Thus, the statement "The range is all real numbers" is False.
4. The range is [tex]\( y > 0 \)[/tex]:
- As mentioned above, [tex]\( f(x) = b^x \)[/tex] produces positive values for all [tex]\( x \)[/tex]. So the range is indeed [tex]\( y > 0 \)[/tex].
- Therefore, the statement "The range is [tex]\( y > 0 \)[/tex]" is True.
5. The graph has an [tex]\( x \)[/tex]-intercept of 1:
- The [tex]\( x \)[/tex]-intercept of a function is the point where the graph crosses the [tex]\( x \)[/tex]-axis, which would happen when [tex]\( y = 0 \)[/tex].
- For [tex]\( f(x) = b^x \)[/tex] where [tex]\( 0 < b < 1 \)[/tex], the output is always positive and never zero, hence there is no [tex]\( x \)[/tex]-intercept.
- Therefore, the statement "The graph has an [tex]\( x \)[/tex]-intercept of 1" is False.
6. The graph has a [tex]\( y \)[/tex]-intercept of 1:
- The [tex]\( y \)[/tex]-intercept of a function is the point where the graph crosses the [tex]\( y \)[/tex]-axis, which happens when [tex]\( x = 0 \)[/tex].
- For [tex]\( f(x) = b^x \)[/tex], substituting [tex]\( x = 0 \)[/tex] gives [tex]\( f(0) = b^0 = 1 \)[/tex].
- Therefore, the statement "The graph has a [tex]\( y \)[/tex]-intercept of 1" is True.
7. The function is always increasing:
- For [tex]\( f(x) = b^x \)[/tex] where [tex]\( 0 < b < 1 \)[/tex], as [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] decreases.
- Therefore, the statement "The function is always increasing" is False.
8. The function is always decreasing:
- As stated above, for [tex]\( 0 < b < 1 \)[/tex], the function [tex]\( f(x) = b^x \)[/tex] decreases as [tex]\( x \)[/tex] increases.
- Therefore, the statement "The function is always decreasing" is True.
Thus, we compile the following verified truths:
1. True
2. False
3. False
4. True
5. False
6. True
7. False
8. True
The statements that are true for [tex]\( f(x)=b^x \)[/tex] where [tex]\( 0- 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.
1. The domain is all real numbers:
- For the function [tex]\( f(x) = b^x \)[/tex] where [tex]\( 0 < b < 1 \)[/tex], [tex]\( x \)[/tex] can take any real value. There are no restrictions on [tex]\( x \)[/tex].
- Therefore, the statement "The domain is all real numbers" is True.
2. The domain is [tex]\( x > 0 \)[/tex]:
- As determined above, the domain of [tex]\( f(x) = b^x \)[/tex] includes all real numbers, not just [tex]\( x > 0 \)[/tex].
- Therefore, the statement "The domain is [tex]\( x > 0 \)[/tex]" is False.
3. The range is all real numbers:
- For [tex]\( f(x) = b^x \)[/tex] where [tex]\( 0 < b < 1 \)[/tex], the output (range) of [tex]\( f(x) \)[/tex] is always positive and never zero or negative.
- Therefore, the range is not all real numbers.
- Thus, the statement "The range is all real numbers" is False.
4. The range is [tex]\( y > 0 \)[/tex]:
- As mentioned above, [tex]\( f(x) = b^x \)[/tex] produces positive values for all [tex]\( x \)[/tex]. So the range is indeed [tex]\( y > 0 \)[/tex].
- Therefore, the statement "The range is [tex]\( y > 0 \)[/tex]" is True.
5. The graph has an [tex]\( x \)[/tex]-intercept of 1:
- The [tex]\( x \)[/tex]-intercept of a function is the point where the graph crosses the [tex]\( x \)[/tex]-axis, which would happen when [tex]\( y = 0 \)[/tex].
- For [tex]\( f(x) = b^x \)[/tex] where [tex]\( 0 < b < 1 \)[/tex], the output is always positive and never zero, hence there is no [tex]\( x \)[/tex]-intercept.
- Therefore, the statement "The graph has an [tex]\( x \)[/tex]-intercept of 1" is False.
6. The graph has a [tex]\( y \)[/tex]-intercept of 1:
- The [tex]\( y \)[/tex]-intercept of a function is the point where the graph crosses the [tex]\( y \)[/tex]-axis, which happens when [tex]\( x = 0 \)[/tex].
- For [tex]\( f(x) = b^x \)[/tex], substituting [tex]\( x = 0 \)[/tex] gives [tex]\( f(0) = b^0 = 1 \)[/tex].
- Therefore, the statement "The graph has a [tex]\( y \)[/tex]-intercept of 1" is True.
7. The function is always increasing:
- For [tex]\( f(x) = b^x \)[/tex] where [tex]\( 0 < b < 1 \)[/tex], as [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] decreases.
- Therefore, the statement "The function is always increasing" is False.
8. The function is always decreasing:
- As stated above, for [tex]\( 0 < b < 1 \)[/tex], the function [tex]\( f(x) = b^x \)[/tex] decreases as [tex]\( x \)[/tex] increases.
- Therefore, the statement "The function is always decreasing" is True.
Thus, we compile the following verified truths:
1. True
2. False
3. False
4. True
5. False
6. True
7. False
8. True
The statements that are true for [tex]\( f(x)=b^x \)[/tex] where [tex]\( 0- 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.
