To evaluate [tex]\((m \circ n)(4)\)[/tex], we must first determine the composition of the functions [tex]\( m \)[/tex] and [tex]\( n \)[/tex].
We are given:
[tex]\[
m(x) = 4x - 5
\][/tex]
[tex]\[
n(x) = 2x + 11
\][/tex]
The composition [tex]\((m \circ n)(x)\)[/tex] means substituting [tex]\( n(x) \)[/tex] into [tex]\( m(x) \)[/tex]. This can be written as:
[tex]\[
(m \circ n)(x) = m(n(x))
\][/tex]
Next, we substitute [tex]\( n(x) \)[/tex] into [tex]\( m(x) \)[/tex]:
[tex]\[
m(2x + 11) = 4(2x + 11) - 5
\][/tex]
Performing the multiplication and simplification, we get:
[tex]\[
m(2x + 11) = 4 \cdot 2x + 4 \cdot 11 - 5
\][/tex]
[tex]\[
m(2x + 11) = 8x + 44 - 5
\][/tex]
[tex]\[
m(2x + 11) = 8x + 39
\][/tex]
Now, to evaluate [tex]\((m \circ n)(4)\)[/tex], substitute [tex]\( x = 4 \)[/tex] in [tex]\( m(2x + 11) \)[/tex]:
[tex]\[
(m \circ n)(4) = 8(4) + 39
\][/tex]
[tex]\[
(m \circ n)(4) = 32 + 39
\][/tex]
[tex]\[
(m \circ n)(4) = 71
\][/tex]
We then compare this result with the given options to see if any match:
A. [tex]\((m n)(4) = 8(4)^2 - 6(4) - 55\)[/tex]
B. [tex]\((m n)(4) = 8(4)^2 - 55\)[/tex]
C. [tex]\((m n)(4) = 8(4)^2 + 34(4) - 55\)[/tex]
D. [tex]\((m n)(4) = 8(4) - 55\)[/tex]
Evaluating each:
Option A:
[tex]\[
8(4)^2 - 6(4) - 55 = 8(16) - 24 - 55 = 128 - 79 = 49
\][/tex]
Option B:
[tex]\[
8(4)^2 - 55 = 8(16) - 55 = 128 - 55 = 73
\][/tex]
Option C:
[tex]\[
8(4)^2 + 34(4) - 55 = 8(16) + 136 - 55 = 128 + 136 - 55 = 264 - 55 = 209
\][/tex]
Option D:
[tex]\[
8(4) - 55 = 32 - 55 = -23
\][/tex]
None of the provided options match the correct evaluation of 71. Therefore, the correct answer is:
[tex]\[
\boxed{\text{No correct option available}}
\][/tex]
