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