To determine which statement is true for the function [tex]\( f(x) = \left(\frac{9}{10}\right)^x \)[/tex], let's analyze each statement carefully.
### Statement A: The domain of [tex]\( f(x) \)[/tex] is [tex]\( x > 0 \)[/tex].
- The domain of [tex]\( f(x) = \left(\frac{9}{10}\right)^x \)[/tex] is the set of all real numbers. This is because you can plug any real number into [tex]\( x \)[/tex] and the function will produce a valid output.
- Conclusion: The domain is not limited to [tex]\( x > 0 \)[/tex]; it includes all real numbers. Therefore, statement A is false.
### Statement B: The [tex]\( y \)[/tex]-intercept is [tex]\((0,1)\)[/tex].
- To find the [tex]\( y \)[/tex]-intercept, we substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = \left(\frac{9}{10}\right)^0 = 1. \][/tex]
- Therefore, the [tex]\( y \)[/tex]-intercept is [tex]\((0,1)\)[/tex].
- Conclusion: Statement B is true.
### Statement C: The range of [tex]\( f(x) \)[/tex] is [tex]\( y > \frac{9}{10} \)[/tex].
- For an exponential function of the form [tex]\( f(x) = a^x \)[/tex] where [tex]\( 0 < a < 1 \)[/tex] (in this case, [tex]\(\frac{9}{10}\)[/tex]), the range is [tex]\( 0 < y < 1 \)[/tex].
- Therefore, [tex]\( f(x) \)[/tex] can take any value between 0 and 1, but never actually reaches 0 or 1.
- Conclusion: The range is [tex]\( 0 < y < 1 \)[/tex], so statement C is false.
### Statement D: It is always increasing.
- For an exponential function of the form [tex]\( f(x) = a^x \)[/tex] where [tex]\( 0 < a < 1 \)[/tex] (in this case, [tex]\(\frac{9}{10}\)[/tex]), the function is always decreasing because [tex]\( \left(\frac{9}{10}\right)^x \)[/tex] gets smaller as [tex]\( x \)[/tex] gets larger.
- Conclusion: The function is decreasing, not increasing. Therefore, statement D is false.
From the analysis above, the only true statement is:
B. The [tex]\( y \)[/tex]-intercept is [tex]\((0,1)\)[/tex].
