To determine the range of the exponential function [tex]\( p(x) \)[/tex] that increases at a rate of [tex]\( 25\% \)[/tex], given it includes the point [tex]\((0,10)\)[/tex] and is shifted down 5 units, follow these steps:
1. Base Function Analysis:
The exponential function that increases at a rate of [tex]\( 25\% \)[/tex] can be written as [tex]\( f(x) = 10 \cdot 1.25^x \)[/tex], where [tex]\( f(0) = 10 \)[/tex].
2. Vertical Shift:
If we shift the function down by 5 units, the new function becomes [tex]\( g(x) = 10 \cdot 1.25^x - 5 \)[/tex].
3. Determining the Range:
Original exponential functions of the form [tex]\( 10 \cdot 1.25^x \)[/tex] have a range of [tex]\((0, \infty)\)[/tex]. When we subtract 5 from the output, the range shifts downward by 5 units.
Therefore, the range of our transformed function [tex]\( g(x) = 10 \cdot 1.25^x - 5 \)[/tex], changes accordingly:
[tex]\[
\text{Range of } g(x) = \left(0 - 5, \infty - 5\right)
\][/tex]
[tex]\[
\text{Range of } g(x) = (-5, \infty)
\][/tex]
Thus, the range of the function [tex]\( p(x) \)[/tex] is [tex]\( (-5, \infty) \)[/tex].
Therefore, the correct option is:
[tex]\[
(-5, \infty)
\][/tex]
