Let's find the value of the piecewise function [tex]\( p(x) \)[/tex] for different values of [tex]\( x \)[/tex].
The function [tex]\( p(x) \)[/tex] is defined as:
[tex]\[
p(x) = \begin{cases}
x - 1, & \text{if } x \leq -2 \\
2x - 1, & \text{if } x > -2
\end{cases}
\][/tex]
a. To find [tex]\( p(-20) \)[/tex]:
Since [tex]\(-20 \leq -2\)[/tex], we use the first piece of the function:
[tex]\[ p(-20) = -20 - 1 = -21 \][/tex]
b. To find [tex]\( p(-2) \)[/tex]:
Since [tex]\(-2 \leq -2\)[/tex], we use the first piece of the function:
[tex]\[ p(-2) = -2 - 1 = -3 \][/tex]
c. To find [tex]\( p(4) \)[/tex]:
Since [tex]\( 4 > -2 \)[/tex], we use the second piece of the function:
[tex]\[ p(4) = 2 \cdot 4 - 1 = 8 - 1 = 7 \][/tex]
d. To find [tex]\( p(5.7) \)[/tex]:
Since [tex]\( 5.7 > -2 \)[/tex], we use the second piece of the function:
[tex]\[ p(5.7) = 2 \cdot 5.7 - 1 = 11.4 - 1 = 10.4 \][/tex]
e. To find [tex]\( p(-2) + p(4) \)[/tex]:
We have already found [tex]\( p(-2) = -3 \)[/tex] and [tex]\( p(4) = 7 \)[/tex].
Adding these values together:
[tex]\[ p(-2) + p(4) = -3 + 7 = 4 \][/tex]
The results are:
a. [tex]\( p(-20) = -21 \)[/tex]
b. [tex]\( p(-2) = -3 \)[/tex]
c. [tex]\( p(4) = 7 \)[/tex]
d. [tex]\( p(5.7) = 10.4 \)[/tex]
e. [tex]\( p(-2) + p(4) = 4 \)[/tex]
