Answer :
Let's go through the problem step-by-step to find and simplify each of the given composite functions:
### a. [tex]\((d \circ a)(x)\)[/tex]
To find [tex]\((d \circ a)(x)\)[/tex], we need to evaluate the function [tex]\(d\)[/tex] at the value given by [tex]\(a(x)\)[/tex]:
1. Define [tex]\(a(x)\)[/tex]: [tex]\(a(x) = x + 2\)[/tex]
2. Substitute [tex]\(a(x)\)[/tex] into [tex]\(d(x)\)[/tex]:
[tex]\[ (d \circ a)(x) = d(a(x)) = d(x + 2) \][/tex]
3. Define [tex]\(d(x)\)[/tex]: [tex]\(d(x) = \sqrt{x + 5}\)[/tex]
4. Substitute [tex]\(x + 2\)[/tex] into [tex]\(d(x)\)[/tex]:
[tex]\[ d(x + 2) = \sqrt{(x + 2) + 5} = \sqrt{x + 7} \][/tex]
Thus, the simplified form of [tex]\((d \circ a)(x)\)[/tex] is:
[tex]\[ (d \circ a)(x) = \sqrt{x + 7} \][/tex]
### b. [tex]\((d \circ a)(2)\)[/tex]
To find [tex]\((d \circ a)(2)\)[/tex], we use the simplified form from part (a) and substitute [tex]\(x = 2\)[/tex]:
1. Use the result from (a):
[tex]\[ (d \circ a)(x) = \sqrt{x + 7} \][/tex]
2. Substitute [tex]\(x = 2\)[/tex]:
[tex]\[ (d \circ a)(2) = \sqrt{2 + 7} = \sqrt{9} = 3 \][/tex]
So, [tex]\((d \circ a)(2) = 3\)[/tex].
### c. [tex]\((c \circ m)(x)\)[/tex]
To find [tex]\((c \circ m)(x)\)[/tex], we need to evaluate the function [tex]\(c\)[/tex] at the value given by [tex]\(m(x)\)[/tex]:
1. Define [tex]\(m(x)\)[/tex]: [tex]\(m(x) = 5x - 3\)[/tex]
2. Substitute [tex]\(m(x)\)[/tex] into [tex]\(c(x)\)[/tex]:
[tex]\[ (c \circ m)(x) = c(m(x)) = c(5x - 3) \][/tex]
3. Define [tex]\(c(x)\)[/tex]: [tex]\(c(x) = 5x^2 + 17x - 12\)[/tex]
4. Substitute [tex]\(5x - 3\)[/tex] into [tex]\(c(x)\)[/tex]:
[tex]\[ c(5x - 3) = 5(5x - 3)^2 + 17(5x - 3) - 12 \][/tex]
5. Simplify the expression:
[tex]\[ (5x - 3)^2 = 25x^2 - 30x + 9 \][/tex]
[tex]\[ c(5x - 3) = 5(25x^2 - 30x + 9) + 17(5x - 3) - 12 \][/tex]
[tex]\[ = 125x^2 - 150x + 45 + 85x - 51 - 12 \][/tex]
[tex]\[ = 125x^2 - 65x - 18 \][/tex]
Thus, the simplified form of [tex]\((c \circ m)(x)\)[/tex] is:
[tex]\[ (c \circ m)(x) = 125x^2 - 65x - 18 \][/tex]
### d. [tex]\((m \circ c)(x)\)[/tex]
To find [tex]\((m \circ c)(x)\)[/tex], we need to evaluate the function [tex]\(m\)[/tex] at the value given by [tex]\(c(x)\)[/tex]:
1. Define [tex]\(c(x)\)[/tex]: [tex]\(c(x) = 5x^2 + 17x - 12\)[/tex]
2. Substitute [tex]\(c(x)\)[/tex] into [tex]\(m(x)\)[/tex]:
[tex]\[ (m \circ c)(x) = m(c(x)) = m(5x^2 + 17x - 12) \][/tex]
3. Define [tex]\(m(x)\)[/tex]: [tex]\(m(x) = 5x - 3\)[/tex]
4. 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]
5. Simplify the expression:
[tex]\[ = 25x^2 + 85x - 60 - 3 \][/tex]
[tex]\[ = 25x^2 + 85x - 63 \][/tex]
Thus, the simplified form of [tex]\((m \circ c)(x)\)[/tex] is:
[tex]\[ (m \circ c)(x) = 25x^2 + 85x - 63 \][/tex]
### Summary
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]
### a. [tex]\((d \circ a)(x)\)[/tex]
To find [tex]\((d \circ a)(x)\)[/tex], we need to evaluate the function [tex]\(d\)[/tex] at the value given by [tex]\(a(x)\)[/tex]:
1. Define [tex]\(a(x)\)[/tex]: [tex]\(a(x) = x + 2\)[/tex]
2. Substitute [tex]\(a(x)\)[/tex] into [tex]\(d(x)\)[/tex]:
[tex]\[ (d \circ a)(x) = d(a(x)) = d(x + 2) \][/tex]
3. Define [tex]\(d(x)\)[/tex]: [tex]\(d(x) = \sqrt{x + 5}\)[/tex]
4. Substitute [tex]\(x + 2\)[/tex] into [tex]\(d(x)\)[/tex]:
[tex]\[ d(x + 2) = \sqrt{(x + 2) + 5} = \sqrt{x + 7} \][/tex]
Thus, the simplified form of [tex]\((d \circ a)(x)\)[/tex] is:
[tex]\[ (d \circ a)(x) = \sqrt{x + 7} \][/tex]
### b. [tex]\((d \circ a)(2)\)[/tex]
To find [tex]\((d \circ a)(2)\)[/tex], we use the simplified form from part (a) and substitute [tex]\(x = 2\)[/tex]:
1. Use the result from (a):
[tex]\[ (d \circ a)(x) = \sqrt{x + 7} \][/tex]
2. Substitute [tex]\(x = 2\)[/tex]:
[tex]\[ (d \circ a)(2) = \sqrt{2 + 7} = \sqrt{9} = 3 \][/tex]
So, [tex]\((d \circ a)(2) = 3\)[/tex].
### c. [tex]\((c \circ m)(x)\)[/tex]
To find [tex]\((c \circ m)(x)\)[/tex], we need to evaluate the function [tex]\(c\)[/tex] at the value given by [tex]\(m(x)\)[/tex]:
1. Define [tex]\(m(x)\)[/tex]: [tex]\(m(x) = 5x - 3\)[/tex]
2. Substitute [tex]\(m(x)\)[/tex] into [tex]\(c(x)\)[/tex]:
[tex]\[ (c \circ m)(x) = c(m(x)) = c(5x - 3) \][/tex]
3. Define [tex]\(c(x)\)[/tex]: [tex]\(c(x) = 5x^2 + 17x - 12\)[/tex]
4. Substitute [tex]\(5x - 3\)[/tex] into [tex]\(c(x)\)[/tex]:
[tex]\[ c(5x - 3) = 5(5x - 3)^2 + 17(5x - 3) - 12 \][/tex]
5. Simplify the expression:
[tex]\[ (5x - 3)^2 = 25x^2 - 30x + 9 \][/tex]
[tex]\[ c(5x - 3) = 5(25x^2 - 30x + 9) + 17(5x - 3) - 12 \][/tex]
[tex]\[ = 125x^2 - 150x + 45 + 85x - 51 - 12 \][/tex]
[tex]\[ = 125x^2 - 65x - 18 \][/tex]
Thus, the simplified form of [tex]\((c \circ m)(x)\)[/tex] is:
[tex]\[ (c \circ m)(x) = 125x^2 - 65x - 18 \][/tex]
### d. [tex]\((m \circ c)(x)\)[/tex]
To find [tex]\((m \circ c)(x)\)[/tex], we need to evaluate the function [tex]\(m\)[/tex] at the value given by [tex]\(c(x)\)[/tex]:
1. Define [tex]\(c(x)\)[/tex]: [tex]\(c(x) = 5x^2 + 17x - 12\)[/tex]
2. Substitute [tex]\(c(x)\)[/tex] into [tex]\(m(x)\)[/tex]:
[tex]\[ (m \circ c)(x) = m(c(x)) = m(5x^2 + 17x - 12) \][/tex]
3. Define [tex]\(m(x)\)[/tex]: [tex]\(m(x) = 5x - 3\)[/tex]
4. 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]
5. Simplify the expression:
[tex]\[ = 25x^2 + 85x - 60 - 3 \][/tex]
[tex]\[ = 25x^2 + 85x - 63 \][/tex]
Thus, the simplified form of [tex]\((m \circ c)(x)\)[/tex] is:
[tex]\[ (m \circ c)(x) = 25x^2 + 85x - 63 \][/tex]
### Summary
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]
