To determine the range of the piecewise function [tex]\( g(x) \)[/tex], we need to analyze each part of the function separately and then combine the results.
The function is defined as:
[tex]\[ g(x) = \begin{cases}
x^2 - 5 & \text{if } x < -3 \\
2x & \text{if } x \geq -3
\end{cases} \][/tex]
1. Analyzing [tex]\( g(x) = x^2 - 5 \)[/tex] for [tex]\( x < -3 \)[/tex]:
- The function [tex]\( g(x) = x^2 - 5 \)[/tex] is a quadratic equation.
- The quadratic equation [tex]\( x^2 - 5 \)[/tex] represents a parabola opening upwards since the coefficient of [tex]\( x^2 \)[/tex] is positive.
- As [tex]\( x \)[/tex] becomes more negative (i.e., towards [tex]\(-\infty\)[/tex]), [tex]\( x^2 \)[/tex] increases and thus [tex]\( x^2 - 5 \)[/tex] also increases.
- To find how low [tex]\( g(x) = x^2 - 5 \)[/tex] can go, we examine it at [tex]\( x = -3 \)[/tex], the boundary of the interval:
[tex]\[ g(-3) = (-3)^2 - 5 = 9 - 5 = 4 \][/tex]
- Therefore, for [tex]\( x < -3 \)[/tex], [tex]\( g(x) \)[/tex] can take any value less than [tex]\( 4 \)[/tex]. This gives us the range from [tex]\( (-\infty, 4) \)[/tex].
2. Analyzing [tex]\( g(x) = 2x \)[/tex] for [tex]\( x \geq -3 \)[/tex]:
- The function [tex]\( g(x) = 2x \)[/tex] is a linear equation with a positive slope.
- As [tex]\( x \)[/tex] becomes more positive (i.e., towards [tex]\( \infty \)[/tex]), [tex]\( 2x \)[/tex] increases without bound.
- To find the minimum value of [tex]\( g(x) = 2x \)[/tex] in this interval, we examine it at [tex]\( x = -3 \)[/tex], the boundary of the interval:
[tex]\[ g(-3) = 2(-3) = -6 \][/tex]
- Therefore, for [tex]\( x \geq -3 \)[/tex], [tex]\( g(x) \)[/tex] can take any value greater than or equal to [tex]\( -6 \)[/tex]. This gives us the range [tex]\( [-6, \infty) \)[/tex].
3. Combining the ranges:
- From the analysis above, for [tex]\( x < -3 \)[/tex], the range is [tex]\( (-\infty, 4) \)[/tex].
- For [tex]\( x \geq -3 \)[/tex], the range is [tex]\( [-6, \infty) \)[/tex].
- When we merge these two intervals, we need to combine [tex]\( (-\infty, 4) \)[/tex] with [tex]\( [-6, \infty) \)[/tex].
Since the intervals overlap, particularly because [tex]\([-6, 4)\)[/tex] fits within both intervals, and when considering the nature of the function at [tex]\( x = -3 \)[/tex], where [tex]\( 4 > -6 \)[/tex] doesn't affect the continuity of the domain, the range of the combined function is:
[tex]\[ (-\infty, \infty) \][/tex]
Therefore, the range of [tex]\( g(x) \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].
