Certainly! Let’s solve the problem step-by-step.
First, we need to find the composition functions for the given functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
Given:
[tex]\[ f(x) = 5x^2 - 2x + 4 \][/tex]
[tex]\[
g(x) = \begin{cases}
6x - 1 & \text{if } x < 2 \\
4 & \text{if } 2 \leq x < 7 \\
x - 2 & \text{if } x \geq 7
\end{cases}
\][/tex]
Part (a): [tex]\((f \circ g)(6)\)[/tex]
This part asks us to find [tex]\( (f \circ g)(6) \)[/tex], which means [tex]\( f(g(6)) \)[/tex].
1. Determine [tex]\( g(6) \)[/tex]:
- Since [tex]\( 2 \leq 6 < 7 \)[/tex], we use the second case of [tex]\( g(x) \)[/tex]: [tex]\( g(6) = 4 \)[/tex].
2. Now, compute [tex]\( f(g(6)) = f(4) \)[/tex]:
- Substitute [tex]\( x = 4 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(4) = 5(4)^2 - 2(4) + 4
\][/tex]
- Simplify:
[tex]\[
f(4) = 5(16) - 8 + 4 = 80 - 8 + 4 = 76
\][/tex]
Thus,
[tex]\[ (f \circ g)(6) = 76 \][/tex]
Part (b): [tex]\((g \circ f)(1)\)[/tex]
This part asks us to find [tex]\( (g \circ f)(1) \)[/tex], which means [tex]\( g(f(1)) \)[/tex].
1. Determine [tex]\( f(1) \)[/tex]:
- Substitute [tex]\( x = 1 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(1) = 5(1)^2 - 2(1) + 4
\][/tex]
- Simplify:
[tex]\[
f(1) = 5(1) - 2 + 4 = 5 - 2 + 4 = 7
\][/tex]
2. Now, compute [tex]\( g(f(1)) = g(7) \)[/tex]:
- Since [tex]\( 7 \geq 7 \)[/tex], we use the third case of [tex]\( g(x) \)[/tex]: [tex]\( g(7) = 7 - 2 \)[/tex].
[tex]\[
g(7) = 5
\][/tex]
Thus,
[tex]\[ (g \circ f)(1) = 5 \][/tex]
Final results:
(a) [tex]\( (f \circ g)(6) = 76 \)[/tex]
(b) [tex]\( (g \circ f)(1) = 5 \)[/tex]
Therefore, we have:
[tex]\[
\boxed{(f \circ g)(6) = 76 \quad \text{and} \quad (g \circ f)(1) = 5}
\][/tex]
