Question
\( \left. \begin{array} { l } { f ( x ) = 2 x - 9 } \\ { g ( x ) = - 5 x + 6 } \\ { ( f \circ g ) ( x ) = f ( g x ) ) } \end{array} \right. \)
Ask by Burton French. in the United States
Mar 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( (f \circ g)(x) = -10x + 3 \)
Solution
To find the composition of the functions \( f \) and \( g \), denoted as \( (f \circ g)(x) \), we need to substitute \( g(x) \) into \( f(x) \).
1. **Identify the functions:**
- \( f(x) = 2x - 9 \)
- \( g(x) = -5x + 6 \)
2. **Substitute \( g(x) \) into \( f(x) \):**
\[
(f \circ g)(x) = f(g(x)) = f(-5x + 6)
\]
3. **Calculate \( f(-5x + 6) \):**
Substitute \( -5x + 6 \) into the function \( f \):
\[
f(-5x + 6) = 2(-5x + 6) - 9
\]
4. **Distribute and simplify:**
\[
= 2(-5x) + 2(6) - 9
\]
\[
= -10x + 12 - 9
\]
\[
= -10x + 3
\]
Thus, the composition of the functions is:
\[
(f \circ g)(x) = -10x + 3
\]
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Simplify this solution Beyond the Answer
To find \( (f \circ g)(x) \), you first need to substitute \( g(x) \) into \( f(x) \). So, calculate \( g(x) \) first: \( g(x) = -5x + 6 \). Now plug this into \( f(x) \): \( f(g(x)) = f(-5x + 6) = 2(-5x + 6) - 9 \). Simplifying that gives: \( f(g(x)) = -10x + 12 - 9 = -10x + 3 \). Thus, \( (f \circ g)(x) = -10x + 3 \). Now, if you want to take it a step further, what would happen if you wanted to find \( (g \circ f)(x) \)? It's like a dance move between functions! You simply switch the roles: take \( f(x) \) and substitute it into \( g(x) \). This strong synergy between functions can lead to new insights!
