Let's analyze the given functions and solve the problem step-by-step. The functions given are:
[tex]\[ p(x) = -5x - 5 \][/tex]
[tex]\[ q(x) = -2x - 1 \][/tex]
We need to find the values of [tex]\( (p \circ q)(4) \)[/tex] and [tex]\( (q \circ p)(4) \)[/tex].
### Finding [tex]\( (p \circ q)(4) \)[/tex]
1. The composition [tex]\( (p \circ q)(x) \)[/tex] means applying the function [tex]\( q(x) \)[/tex] first, then applying the function [tex]\( p(x) \)[/tex] to the result.
2. First, calculate [tex]\( q(4) \)[/tex]:
[tex]\[ q(4) = -2(4) - 1 = -8 - 1 = -9 \][/tex]
3. Next, use this result to find [tex]\( p(-9) \)[/tex]:
[tex]\[ p(-9) = -5(-9) - 5 = 45 - 5 = 40 \][/tex]
Thus, [tex]\( (p \circ q)(4) = 40 \)[/tex].
### Finding [tex]\( (q \circ p)(4) \)[/tex]
1. The composition [tex]\( (q \circ p)(x) \)[/tex] means applying the function [tex]\( p(x) \)[/tex] first, then applying the function [tex]\( q(x) \)[/tex] to the result.
2. First, calculate [tex]\( p(4) \)[/tex]:
[tex]\[ p(4) = -5(4) - 5 = -20 - 5 = -25 \][/tex]
3. Next, use this result to find [tex]\( q(-25) \)[/tex]:
[tex]\[ q(-25) = -2(-25) - 1 = 50 - 1 = 49 \][/tex]
Thus, [tex]\( (q \circ p)(4) = 49 \)[/tex].
The final answers are:
[tex]\[ (p \circ q)(4) = 40 \][/tex]
[tex]\[ (q \circ p)(4) = 49 \][/tex]
