To solve the problem of finding [tex]\( x \)[/tex] such that [tex]\( (m \circ n)(x) = 2 \)[/tex], we need to follow these steps:
1. Understand [tex]\( (m \circ n)(x) = 2 \)[/tex]:
The notation [tex]\( (m \circ n)(x) \)[/tex] means we first apply the function [tex]\( n \)[/tex] to [tex]\( x \)[/tex], and then apply the function [tex]\( m \)[/tex] to the result of [tex]\( n(x) \)[/tex].
2. Simplify the problem:
Since [tex]\( (m \circ n)(x) = 2 \)[/tex], it implies:
[tex]\[
m(n(x)) = 2
\][/tex]
This means we need to find the value of [tex]\( x \)[/tex] such that [tex]\( n(x) \)[/tex] equals a specific value that results in [tex]\( m \)[/tex] producing 2. However, without knowing [tex]\( m \)[/tex], we focus on [tex]\( x \)[/tex] for which [tex]\( n(x) \)[/tex] equals 2 directly.
3. Use the given table to check [tex]\( n(x) \)[/tex]:
We need to examine each [tex]\( x \)[/tex] in the table to find where [tex]\( n(x) \)[/tex] equals 2.
The given table is:
[tex]\[
\begin{array}{cccccc}
x & -5 & -3 & -1 & 3 & 5 \\
n(x) & 2 & 1 & -3 & 1.5 & 0
\end{array}
\][/tex]
4. Find the [tex]\( n(x) \)[/tex] value:
We look through each entry in the row of [tex]\( n(x) \)[/tex]:
- For [tex]\( x = -5 \)[/tex], [tex]\( n(-5) = 2 \)[/tex]
- For [tex]\( x = -3 \)[/tex], [tex]\( n(-3) = 1 \)[/tex]
- For [tex]\( x = -1 \)[/tex], [tex]\( n(-1) = -3 \)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\( n(3) = 1.5 \)[/tex]
- For [tex]\( x = 5 \)[/tex], [tex]\( n(5) = 0 \)[/tex]
5. Identify the correct [tex]\( x \)[/tex]:
We need [tex]\( n(x) = 2 \)[/tex]. From the table, [tex]\( n(-5) = 2 \)[/tex] is the only match.
Therefore, the value of [tex]\( x \)[/tex] when [tex]\( (m \circ n)(x) = 2 \)[/tex] is:
[tex]\[
x = -5
\][/tex]
