Answer :
Let's tackle each part of the question step-by-step, using the mathematical expressions given for [tex]\( d(x) \)[/tex], [tex]\( m(x) \)[/tex], [tex]\( c(x) \)[/tex], and [tex]\( a(x) \)[/tex].
### Part a: [tex]\((d \circ a)(x)\)[/tex]
This denotes the composition of the function [tex]\( d \)[/tex] with [tex]\( a \)[/tex]. It means we first apply [tex]\( a(x) \)[/tex] and then apply [tex]\( d \)[/tex] to the result:
[tex]\[ (d \circ a)(x) = d(a(x)) \][/tex]
Given [tex]\( a(x) = x + 2 \)[/tex], we substitute [tex]\( a(x) \)[/tex] into [tex]\( d(x) \)[/tex]:
[tex]\[ d(a(x)) = d(x + 2) \][/tex]
Since [tex]\( d(x) = \sqrt{x + 5} \)[/tex], we substitute [tex]\( x + 2 \)[/tex] for [tex]\( x \)[/tex] in [tex]\( d(x) \)[/tex]:
[tex]\[ d(x + 2) = \sqrt{(x + 2) + 5} = \sqrt{x + 7} \][/tex]
Therefore,
[tex]\[ (d \circ a)(x) = \sqrt{x + 7} \][/tex]
### Part b: [tex]\((d \circ a)(2)\)[/tex]
We already know from part a that:
[tex]\[ (d \circ a)(x) = \sqrt{x + 7} \][/tex]
Now we need to evaluate this at [tex]\( x = 2 \)[/tex]:
[tex]\[ (d \circ a)(2) = \sqrt{2 + 7} = \sqrt{9} = 3 \][/tex]
Therefore,
[tex]\[ (d \circ a)(2) = 3 \][/tex]
### Part c: [tex]\((c \circ m)(x)\)[/tex]
This denotes the composition of the function [tex]\( c \)[/tex] with [tex]\( m \)[/tex]. It means we first apply [tex]\( m(x) \)[/tex] and then apply [tex]\( c \)[/tex] to the result:
[tex]\[ (c \circ m)(x) = c(m(x)) \][/tex]
Given [tex]\( m(x) = 5x - 3 \)[/tex], we substitute [tex]\( m(x) \)[/tex] into [tex]\( c(x) \)[/tex]:
[tex]\[ c(m(x)) = c(5x - 3) \][/tex]
Since [tex]\( c(x) = 5x^2 + 17x - 12 \)[/tex], we substitute [tex]\( 5x - 3 \)[/tex] for [tex]\( x \)[/tex] in [tex]\( c(x) \)[/tex]:
[tex]\[ c(5x - 3) = 5(5x - 3)^2 + 17(5x - 3) - 12 \][/tex]
To simplify this, first calculate [tex]\( (5x - 3)^2 \)[/tex]:
[tex]\[ (5x - 3)^2 = 25x^2 - 30x + 9 \][/tex]
Then:
[tex]\[ 5(25x^2 - 30x + 9) = 125x^2 - 150x + 45 \][/tex]
Next, distribute 17 over [tex]\( 5x - 3 \)[/tex]:
[tex]\[ 17(5x - 3) = 85x - 51 \][/tex]
Combine these results:
[tex]\[ 125x^2 - 150x + 45 + 85x - 51 - 12 = 125x^2 - 65x - 18 \][/tex]
Therefore,
[tex]\[ (c \circ m)(x) = 125x^2 - 65x - 18 \][/tex]
### Part d: [tex]\((m \circ c)(x)\)[/tex]
This denotes the composition of the function [tex]\( m \)[/tex] with [tex]\( c \)[/tex]. It means we first apply [tex]\( c(x) \)[/tex] and then apply [tex]\( m \)[/tex] to the result:
[tex]\[ (m \circ c)(x) = m(c(x)) \][/tex]
Given [tex]\( c(x) = 5x^2 + 17x - 12 \)[/tex], we substitute [tex]\( c(x) \)[/tex] into [tex]\( m(x) \)[/tex]:
[tex]\[ m(c(x)) = m(5x^2 + 17x - 12) \][/tex]
Since [tex]\( m(x) = 5x - 3 \)[/tex], we substitute [tex]\( 5x^2 + 17x - 12 \)[/tex] for [tex]\( x \)[/tex] in [tex]\( m(x) \)[/tex]:
[tex]\[ m(5x^2 + 17x - 12) = 5(5x^2 + 17x - 12) - 3 \][/tex]
Distribute 5 over [tex]\( 5x^2 + 17x - 12 \)[/tex]:
[tex]\[ 5(5x^2 + 17x - 12) = 25x^2 + 85x - 60 \][/tex]
Then:
[tex]\[ 25x^2 + 85x - 60 - 3 = 25x^2 + 85x - 63 \][/tex]
Therefore,
[tex]\[ (m \circ c)(x) = 25x^2 + 85x - 63 \][/tex]
### Summary
To summarize our 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]
### Part a: [tex]\((d \circ a)(x)\)[/tex]
This denotes the composition of the function [tex]\( d \)[/tex] with [tex]\( a \)[/tex]. It means we first apply [tex]\( a(x) \)[/tex] and then apply [tex]\( d \)[/tex] to the result:
[tex]\[ (d \circ a)(x) = d(a(x)) \][/tex]
Given [tex]\( a(x) = x + 2 \)[/tex], we substitute [tex]\( a(x) \)[/tex] into [tex]\( d(x) \)[/tex]:
[tex]\[ d(a(x)) = d(x + 2) \][/tex]
Since [tex]\( d(x) = \sqrt{x + 5} \)[/tex], we substitute [tex]\( x + 2 \)[/tex] for [tex]\( x \)[/tex] in [tex]\( d(x) \)[/tex]:
[tex]\[ d(x + 2) = \sqrt{(x + 2) + 5} = \sqrt{x + 7} \][/tex]
Therefore,
[tex]\[ (d \circ a)(x) = \sqrt{x + 7} \][/tex]
### Part b: [tex]\((d \circ a)(2)\)[/tex]
We already know from part a that:
[tex]\[ (d \circ a)(x) = \sqrt{x + 7} \][/tex]
Now we need to evaluate this at [tex]\( x = 2 \)[/tex]:
[tex]\[ (d \circ a)(2) = \sqrt{2 + 7} = \sqrt{9} = 3 \][/tex]
Therefore,
[tex]\[ (d \circ a)(2) = 3 \][/tex]
### Part c: [tex]\((c \circ m)(x)\)[/tex]
This denotes the composition of the function [tex]\( c \)[/tex] with [tex]\( m \)[/tex]. It means we first apply [tex]\( m(x) \)[/tex] and then apply [tex]\( c \)[/tex] to the result:
[tex]\[ (c \circ m)(x) = c(m(x)) \][/tex]
Given [tex]\( m(x) = 5x - 3 \)[/tex], we substitute [tex]\( m(x) \)[/tex] into [tex]\( c(x) \)[/tex]:
[tex]\[ c(m(x)) = c(5x - 3) \][/tex]
Since [tex]\( c(x) = 5x^2 + 17x - 12 \)[/tex], we substitute [tex]\( 5x - 3 \)[/tex] for [tex]\( x \)[/tex] in [tex]\( c(x) \)[/tex]:
[tex]\[ c(5x - 3) = 5(5x - 3)^2 + 17(5x - 3) - 12 \][/tex]
To simplify this, first calculate [tex]\( (5x - 3)^2 \)[/tex]:
[tex]\[ (5x - 3)^2 = 25x^2 - 30x + 9 \][/tex]
Then:
[tex]\[ 5(25x^2 - 30x + 9) = 125x^2 - 150x + 45 \][/tex]
Next, distribute 17 over [tex]\( 5x - 3 \)[/tex]:
[tex]\[ 17(5x - 3) = 85x - 51 \][/tex]
Combine these results:
[tex]\[ 125x^2 - 150x + 45 + 85x - 51 - 12 = 125x^2 - 65x - 18 \][/tex]
Therefore,
[tex]\[ (c \circ m)(x) = 125x^2 - 65x - 18 \][/tex]
### Part d: [tex]\((m \circ c)(x)\)[/tex]
This denotes the composition of the function [tex]\( m \)[/tex] with [tex]\( c \)[/tex]. It means we first apply [tex]\( c(x) \)[/tex] and then apply [tex]\( m \)[/tex] to the result:
[tex]\[ (m \circ c)(x) = m(c(x)) \][/tex]
Given [tex]\( c(x) = 5x^2 + 17x - 12 \)[/tex], we substitute [tex]\( c(x) \)[/tex] into [tex]\( m(x) \)[/tex]:
[tex]\[ m(c(x)) = m(5x^2 + 17x - 12) \][/tex]
Since [tex]\( m(x) = 5x - 3 \)[/tex], we substitute [tex]\( 5x^2 + 17x - 12 \)[/tex] for [tex]\( x \)[/tex] in [tex]\( m(x) \)[/tex]:
[tex]\[ m(5x^2 + 17x - 12) = 5(5x^2 + 17x - 12) - 3 \][/tex]
Distribute 5 over [tex]\( 5x^2 + 17x - 12 \)[/tex]:
[tex]\[ 5(5x^2 + 17x - 12) = 25x^2 + 85x - 60 \][/tex]
Then:
[tex]\[ 25x^2 + 85x - 60 - 3 = 25x^2 + 85x - 63 \][/tex]
Therefore,
[tex]\[ (m \circ c)(x) = 25x^2 + 85x - 63 \][/tex]
### Summary
To summarize our 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]
