To evaluate the function [tex]\( G(x) \)[/tex] given by
[tex]\[
G(x) = \begin{cases}
x^2 + 2, & \text{if } x \leq 1 \\
|x - 3|, & \text{if } x > 1
\end{cases}
\][/tex]
we need to consider the given values of [tex]\( x \)[/tex] and determine which part of the piecewise function to use.
### (a) Evaluating [tex]\( G(-5) \)[/tex]
For [tex]\( x = -5 \)[/tex]:
- Since [tex]\(-5 \leq 1\)[/tex], we use the first part of the function: [tex]\( G(x) = x^2 + 2 \)[/tex].
Now, substitute [tex]\( x = -5 \)[/tex] into [tex]\( x^2 + 2 \)[/tex]:
[tex]\[
G(-5) = (-5)^2 + 2 = 25 + 2 = 27
\][/tex]
So, [tex]\( G(-5) = 27 \)[/tex].
### (b) Evaluating [tex]\( G(5) \)[/tex]
For [tex]\( x = 5 \)[/tex]:
- Since [tex]\( 5 > 1 \)[/tex], we use the second part of the function: [tex]\( G(x) = |x - 3| \)[/tex].
Now, substitute [tex]\( x = 5 \)[/tex] into [tex]\( |x - 3| \)[/tex]:
[tex]\[
G(5) = |5 - 3| = |2| = 2
\][/tex]
So, [tex]\( G(5) = 2 \)[/tex].
### (c) Evaluating [tex]\( G(1) \)[/tex]
For [tex]\( x = 1 \)[/tex]:
- Since [tex]\( 1 \leq 1 \)[/tex], we use the first part of the function: [tex]\( G(x) = x^2 + 2 \)[/tex].
Now, substitute [tex]\( x = 1 \)[/tex] into [tex]\( x^2 + 2 \)[/tex]:
[tex]\[
G(1) = 1^2 + 2 = 1 + 2 = 3
\][/tex]
So, [tex]\( G(1) = 3 \)[/tex].
In conclusion:
- [tex]\( G(-5) = 27 \)[/tex]
- [tex]\( G(5) = 2 \)[/tex]
- [tex]\( G(1) = 3 \)[/tex]
