To determine the sections of the graph where the given piecewise function [tex]\( f(x) \)[/tex] is decreasing, we need to analyze each part of the piecewise function separately.
The function is defined as:
[tex]\[
f(x)=\left\{\begin{array}{ll}
-(3 x+7) ; & x < -3 \\
2 x^2 - 16 ; & -3 \leq x \leq 3 \\
-\left(2^x - 10\right) ; & x > 3
\end{array}\right.
\][/tex]
1. For [tex]\( x < -3 \)[/tex]: [tex]\( f(x) = -(3x + 7) \)[/tex]
- The expression for the function is linear, and specifically, it is a line with a slope of [tex]\(-3\)[/tex].
- Since the slope is negative, the function is decreasing for all [tex]\( x < -3 \)[/tex].
2. For [tex]\( -3 \leq x \leq 3 \)[/tex]: [tex]\( f(x) = 2x^2 - 16 \)[/tex]
- The function here is a quadratic function. To determine where it is decreasing, we look at its derivative:
[tex]\[
f'(x) = \frac{d}{dx}(2x^2 - 16) = 4x
\][/tex]
- The derivative [tex]\( 4x \)[/tex] tells us the slope of the function. The function is decreasing where the derivative is negative.
- So, we solve [tex]\( 4x < 0 \)[/tex]:
[tex]\[
4x < 0 \implies x < 0
\][/tex]
- Hence, for the interval [tex]\( -3 \leq x \leq 3 \)[/tex], the function is decreasing in the range [tex]\( -3 \leq x < 0 \)[/tex].
3. For [tex]\( x > 3 \)[/tex]: [tex]\( f(x) = -(2^x - 10) \)[/tex]
- This part of the function is an exponential function. To determine where it is decreasing, we analyze its form.
- [tex]\( 2^x \)[/tex] is an increasing function, and subtracting it from 10 and then negating the result will result in a decreasing function for all [tex]\( x > 3 \)[/tex].
Based on this analysis, we can conclude that the function [tex]\( f(x) \)[/tex] is decreasing in the following intervals:
1. [tex]\( (-\infty, -3) \)[/tex]
2. [tex]\( (-3, 0) \)[/tex]
3. [tex]\( (3, \infty) \)[/tex]
Therefore, the sections of the graph where the function is decreasing are:
[tex]\[
(-\infty, -3), (-3, 0), (3, \infty)
\][/tex]
