Select all the correct locations on the graph. Consider the given piecewise function.

[tex]f(x)=\left\{\begin{array}{ll}
-(3 x+7) ; & x\ \textless \ -3 \\
2 x^2-16 ; & -3 \leq x \leq 3 \\
-\left(2^x-10\right) ; & x\ \textgreater \ 3
\end{array}\right.[/tex]

Select the section(s) of the graph where the function is decreasing.

Find the derivative of each piece of the piecewise function.
Determine the intervals where the derivative is negative for each piece.
For x < − 3 , f ′ ( x ) = − 3 , so the function is decreasing.
For − 3 ≤ x ≤ 3 , f ′ ( x ) = 4 x , so the function is decreasing when x < 0 .
For 3"> x > 3 , f ′ ( x ) = − 2 x ln 2 , so the function is decreasing.
Combine the intervals to find where the function is decreasing: ( − ∞ , 0 ) and ( 3 , ∞ ) .
The function is decreasing on the intervals ( − ∞ , 0 ) and ( 3 , ∞ ) .
( − ∞ , 0 ) ∪ ( 3 , ∞ ) ​

Explanation

Problem Analysis We are given a piecewise function and asked to find the intervals where the function is decreasing. To do this, we need to find the derivative of each piece of the function and determine where the derivative is negative.

Piecewise Function Definition The given piecewise function is:


3 \end{array}\right."> f ( x ) = ⎩ ⎨ ⎧ ​ − ( 3 x + 7 ) ; 2 x 2 − 16 ; − ( 2 x − 10 ) ; ​ x < − 3 − 3 ≤ x ≤ 3 x > 3 ​
We will analyze each piece separately.

Derivative for x < -3 For x < − 3 , f ( x ) = − ( 3 x + 7 ) . The derivative is f ′ ( x ) = − 3 . Since − 3 < 0 , the function is decreasing on the interval ( − ∞ , − 3 ) .

Derivative for -3 <= x <= 3 For − 3 ≤ x ≤ 3 , f ( x ) = 2 x 2 − 16 . The derivative is f ′ ( x ) = 4 x . The function is decreasing when 4 x < 0 , which means x < 0 . So the function is decreasing on the interval [ − 3 , 0 ) .

Derivative for x > 3 For 3"> x > 3 , f ( x ) = − ( 2 x − 10 ) . The derivative is f ′ ( x ) = − 2 x ln 2 . Since − 2 x ln 2 < 0 for all x , the function is decreasing on the interval ( 3 , ∞ ) .

Combining Intervals Combining the intervals where the function is decreasing, we have ( − ∞ , − 3 ) , [ − 3 , 0 ) , and ( 3 , ∞ ) . Therefore, the function is decreasing on the intervals ( − ∞ , 0 ) and ( 3 , ∞ ) .

Final Answer The function is decreasing on the intervals ( − ∞ , 0 ) and ( 3 , ∞ ) .


Examples
Understanding where a function is decreasing is crucial in many real-world applications. For example, in economics, you might analyze a cost function to determine when the cost is decreasing as production increases. In physics, you could study the velocity of an object to find when it's slowing down. These concepts help optimize processes and make informed decisions.

Answered by GinnyAnswer | 2025-07-05