To find the equation that defines the function [tex]\( p(x) \)[/tex] given that [tex]\( p(0) = 13 \)[/tex] and the value of [tex]\( p(x) \)[/tex] decreases by 14% for every increase in the value of [tex]\( x \)[/tex] by 1, follow these steps:
1. Understand the Initial Value and Decrease Rate:
- The initial value of the function, when [tex]\( x = 0 \)[/tex], is [tex]\( p(0) = 13 \)[/tex].
- For every increase in [tex]\( x \)[/tex] by 1, [tex]\( p(x) \)[/tex] decreases by 14%. This means that [tex]\( p(x) \)[/tex] becomes 86% of its previous value because 100% - 14% = 86%.
2. Express the Function in Terms of Multiplication:
- If [tex]\( p(x) \)[/tex] decreases to 86% of its value for every 1 unit increase in [tex]\( x \)[/tex], we can express this relationship as:
[tex]\[
p(x) = p(0) \times (0.86)^x
\][/tex]
3. Substitute the Initial Value:
- Substitute [tex]\( p(0) = 13 \)[/tex] into the equation:
[tex]\[
p(x) = 13 \times (0.86)^x
\][/tex]
Thus, the equation that defines the function [tex]\( p \)[/tex] is:
[tex]\[
p(x) = 13 \times (0.86)^x
\][/tex]
This equation indicates that for every increase in [tex]\( x \)[/tex] by 1, the value of [tex]\( p(x) \)[/tex] is multiplied by 0.86, representing the 14% decrease.
