Sure, let’s go through each part step-by-step.
Given functions:
[tex]\[ d(x) = \sqrt{x+5} \][/tex]
[tex]\[ m(x) = 5x - 3 \][/tex]
[tex]\[ a(x) = x + 2 \][/tex]
[tex]\[ c(x) = 5x^2 + 17x - 12 \][/tex]
We are to find and simplify the compositions:
### a. [tex]\((d \circ a)(x) = d(a(x))\)[/tex]
First, we need to substitute [tex]\( a(x) \)[/tex] into [tex]\( d(x) \)[/tex]:
[tex]\[ a(x) = x + 2 \][/tex]
Next, substitute [tex]\( a(x) \)[/tex] into [tex]\( d(x) \)[/tex]:
[tex]\[ d(a(x)) = d(x + 2) = \sqrt{(x + 2) + 5} \][/tex]
Simplify the expression inside the square root:
[tex]\[ \sqrt{x + 2 + 5} = \sqrt{x + 7} \][/tex]
So:
[tex]\[ (d \circ a)(x) = \sqrt{x + 7} \][/tex]
### b. [tex]\((d \circ a)(2) = d(a(2))\)[/tex]
First, find [tex]\( a(2) \)[/tex]:
[tex]\[ a(2) = 2 + 2 = 4 \][/tex]
Next, substitute [tex]\( 4 \)[/tex] into [tex]\( d(x) \)[/tex]:
[tex]\[ d(4) = \sqrt{4 + 5} = \sqrt{9} = 3 \][/tex]
So:
[tex]\[ (d \circ a)(2) = 3 \][/tex]
### c. [tex]\((c \circ m)(x) = c(m(x))\)[/tex]
First, substitute [tex]\( m(x) \)[/tex] into [tex]\( c(x) \)[/tex]:
[tex]\[ m(x) = 5x - 3 \][/tex]
Next, substitute [tex]\( m(x) \)[/tex] into [tex]\( c(x) \)[/tex]:
[tex]\[ c(m(x)) = c(5x - 3) \][/tex]
Substitute [tex]\( 5x - 3 \)[/tex] into [tex]\( c(x) \)[/tex]:
[tex]\[ c(5x - 3) = 5(5x - 3)^2 + 17(5x - 3) - 12 \][/tex]
Now, simplify:
[tex]\[ (5x - 3)^2 = 25x^2 - 30x + 9 \][/tex]
[tex]\[ c(5x - 3) = 5(25x^2 - 30x + 9) + 17(5x - 3) - 12 \][/tex]
Distribute the constants:
[tex]\[ 5(25x^2 - 30x + 9) + 17(5x - 3) - 12 \][/tex]
[tex]\[ = 125x^2 - 150x + 45 + 85x - 51 - 12 \][/tex]
Combine like terms:
[tex]\[ = 125x^2 - 65x - 18 \][/tex]
So:
[tex]\[ (c \circ m)(x) = 125x^2 - 65x - 18 \][/tex]
### d. [tex]\((m \circ c)(x) = m(c(x))\)[/tex]
First, substitute [tex]\( c(x) \)[/tex] into [tex]\( m(x) \)[/tex]:
[tex]\[ c(x) = 5x^2 + 17x - 12 \][/tex]
Next, substitute [tex]\( c(x) \)[/tex] into [tex]\( m(x) \)[/tex]:
[tex]\[ m(c(x)) = m(5x^2 + 17x - 12) \][/tex]
Substitute [tex]\( 5x^2 + 17x - 12 \)[/tex] into [tex]\( m(x) \)[/tex]:
[tex]\[ m(5x^2 + 17x - 12) = 5(5x^2 + 17x - 12) - 3 \][/tex]
Distribute the 5:
[tex]\[ = 25x^2 + 85x - 60 - 3 \][/tex]
Combine like terms:
[tex]\[ = 25x^2 + 85x - 63 \][/tex]
So:
[tex]\[ (m \circ c)(x) = 25x^2 + 85x - 63 \][/tex]
### Summary of results
a. [tex]\((d \circ a)(x) = \sqrt{x + 7}\)[/tex]
b. [tex]\((d \circ a)(2) = 3\)[/tex]
c. [tex]\((c \circ m)(x) = 125x^2 - 65x - 18\)[/tex]
d. [tex]\((m \circ c)(x) = 25x^2 + 85x - 63\)[/tex]
