Step 1 of 3
Correct
following function.
[tex]g(x)=\left\{\begin{array}{cl} \frac{3}{4 x^3} & \text { if } x\ \textless \ 1 \\ -\frac{7}{4} x & \text { if } x\ \textgreater \ 1 \end{array}\right.[/tex]
3: Identify the general shape and direction of the graph of this function on the interval [tex](-\infty, 1)[/tex].
Answer (1)
The function g ( x ) = 4 x 3 3 is analyzed on the interval ( − ∞ , 1 ) .
As x → − ∞ , g ( x ) approaches 0 from the negative side.
As x → 0 − , g ( x ) approaches − ∞ .
As x → 1 − , g ( x ) approaches 4 3 . The function is decreasing on ( − ∞ , 0 ) and ( 0 , 1 ) .
The graph is a decreasing curve approaching 0 as x → − ∞ , approaching − ∞ as x → 0 − , and approaching 4 3 as x → 1 − .
Explanation
Problem Analysis We are asked to identify the general shape and direction of the graph of the function g ( x ) = 4 x 3 3 on the interval ( − ∞ , 1 ) . This function is a rational function.
Behavior as x approaches -infinity Let's analyze the behavior of the function as x approaches − ∞ . As x becomes a very large negative number, x 3 also becomes a very large negative number. Therefore, 4 x 3 3 approaches 0 from the negative side.
Behavior as x approaches 1 from the left Next, let's analyze the behavior of the function as x approaches 1 from the left (i.e., x → 1 − ). As x gets closer to 1, x 3 also gets closer to 1. Therefore, 4 x 3 3 approaches 4 3 from below.
Behavior as x approaches 0 from the left Now, let's analyze the behavior of the function as x approaches 0 from the left (i.e., x → 0 − ). As x gets closer to 0 from the negative side, x 3 also gets closer to 0 from the negative side. Therefore, 4 x 3 3 approaches − ∞ .
Finding the derivative To determine if the function is increasing or decreasing on the interval ( − ∞ , 0 ) and ( 0 , 1 ) , we can find the derivative g ′ ( x ) and check its sign. The derivative of g ( x ) = 4 x 3 3 = 4 3 x − 3 is g ′ ( x ) = 4 3 ∗ ( − 3 ) ∗ x − 4 = − 4 x 4 9 .
Analyzing the sign of the derivative Since x 4 is always positive for x n e 0 , g ′ ( x ) = − 4 x 4 9 is always negative. Therefore, the function is decreasing on the intervals ( − ∞ , 0 ) and ( 0 , 1 ) .
General shape and direction of the graph Based on the above analysis, the graph of the function g ( x ) = 4 x 3 3 on the interval ( − ∞ , 1 ) starts near 0 (from the negative side) as x approaches − ∞ , decreases as x increases, approaches − ∞ as x approaches 0 from the left, and then continues to decrease from − ∞ to 4 3 as x approaches 1 from the left.
Final Answer Therefore, the general shape of the graph on the interval ( − ∞ , 1 ) is a decreasing curve that approaches 0 as x goes to − ∞ , approaches − ∞ as x approaches 0 from the left, and approaches 4 3 as x approaches 1 from the left.
Examples
Understanding the behavior of rational functions like g ( x ) = 4 x 3 3 is crucial in various fields. For instance, in physics, the electric potential due to a point charge is inversely proportional to the distance from the charge. Analyzing how this potential changes as you move closer to or farther from the charge involves understanding the properties of rational functions. Similarly, in economics, cost functions can sometimes be modeled as rational functions, where understanding their behavior helps in optimizing production and minimizing costs.
