To find the range of the piecewise function [tex]\( g(x) \)[/tex], we need to analyze the behavior of each piece of the function over its respective domain. The function is defined as:
[tex]\[ g(x) = \begin{cases}
x^2 - 5, & \text{for } x < -3 \\
2x, & \text{for } x \geq -3
\end{cases} \][/tex]
Let's break this down into two parts and find the range for each:
### Analysis of [tex]\( g(x) = x^2 - 5 \)[/tex] for [tex]\( x < -3 \)[/tex]
1. Behavior as [tex]\( x \to -\infty \)[/tex]:
- As [tex]\( x \)[/tex] decreases without bound (goes to negative infinity), [tex]\( x^2 \)[/tex] will increase without bound (go to positive infinity).
- Therefore, [tex]\( x^2 - 5 \to \infty \)[/tex].
2. Minimum value when [tex]\( x = -3 \)[/tex]:
- Substitute [tex]\( x = -3 \)[/tex]:
[tex]\[
g(-3) = (-3)^2 - 5 = 9 - 5 = 4
\][/tex]
- So, the expression [tex]\( x^2 - 5 \)[/tex] reaches a minimum value of [tex]\( 4 \)[/tex] at [tex]\( x = -3 \)[/tex] within the interval [tex]\( x < -3 \)[/tex].
Hence, for [tex]\( x < -3 \)[/tex], [tex]\( g(x) = x^2 - 5 \)[/tex] ranges from [tex]\( 4 \)[/tex] to [tex]\( \infty \)[/tex].
### Analysis of [tex]\( g(x) = 2x \)[/tex] for [tex]\( x \geq -3 \)[/tex]
1. Behavior as [tex]\( x \to \infty \)[/tex]:
- As [tex]\( x \)[/tex] increases without bound, [tex]\( 2x \)[/tex] will also increase without bound.
- Therefore, [tex]\( 2x \to \infty \)[/tex].
2. Minimum value when [tex]\( x = -3 \)[/tex]:
- Substitute [tex]\( x = -3 \)[/tex]:
[tex]\[
g(-3) = 2(-3) = -6
\][/tex]
- So, the expression [tex]\( 2x \)[/tex] reaches a minimum value of [tex]\( -6 \)[/tex] at [tex]\( x = -3 \)[/tex] within the interval [tex]\( x \geq -3 \)[/tex].
Hence, for [tex]\( x \geq -3 \)[/tex], [tex]\( g(x) = 2x \)[/tex] ranges from [tex]\( -6 \)[/tex] to [tex]\( \infty \)[/tex].
### Combining the Two Parts
From the two separate analyses, we can combine the ranges:
- For [tex]\( x < -3 \)[/tex], the range is from [tex]\( 4 \)[/tex] to [tex]\( \infty \)[/tex].
- For [tex]\( x \geq -3 \)[/tex], the range is from [tex]\( -6 \)[/tex] to [tex]\( \infty \)[/tex].
Thus, the overall range of the piecewise function [tex]\( g(x) \)[/tex] is from the smallest minimum value we found across both pieces. The minimum value across both parts is [tex]\( -6 \)[/tex].
Therefore, the range of [tex]\( g(x) \)[/tex] is:
[tex]\[
[-6, \infty)
\][/tex]
