Problem 104 - Algebra and Trigonometry

Question

In Exercises 104–107, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(f(x)=-x^{3}+4 x,\) then the graph of \(f\) falls to the left and falls to the right.

Step-by-Step Solution

Answer

The statement 'If \(f(x)=-x^{3}+4x\), then the graph of \(f\) falls to the left and falls to the right.' is false. The correct statement should be: 'If \(f(x)=-x^{3}+4 x,\) then the graph of \(f\) rises to the left and falls to the right.'

Step 1: Check If the function is Odd or Even

Since all the powers of \(x\) in the function \(f(x) = -x^{3} + 4x\) are odd and the coefficients are real numbers, it can be concluded that this function is an odd function. Odd functions have the property that \(f(-x) = -f(x)\). This means the graph of an odd function is symmetric with respect to the origin.

Step 2: Analyze the behavior of the function

In a cubic function, the sign of the highest degree term dictates the ends of the graph. Here, the coefficient of \(x^3\) is negative, so the graph 'rises' to the left and 'falls' to the right. This goes contrary to the given statement that the function 'falls' to the left and 'falls' to the right. Hence, the statement is false.

Step 3: Making necessary changes

To make the given statement true, it should be corrected as: 'If \(f(x)=-x^{3}+4x\), then the graph of \(f\) rises to the left and falls to the right.'

Key Concepts

Odd Functions

Odd functions have unique characteristics that make them interesting to study. Simply put, an odd function satisfies the property that \( f(-x) = -f(x) \) for all \( x \) in the function's domain.
This means that if you reflect the graph of an odd function over the x-axis and then again over the y-axis, you'll end up with the original graph. In essence, odd functions are symmetric with respect to the origin.

• Example of an odd function: \( f(x) = -x^3 + 4x \)
• Important property: Symmetry about the origin

This symmetry means if a point \((a, b)\) exists on the graph of an odd function, then \((-a, -b)\) will also lie on the graph. Recognizing odd functions is crucial for understanding the underlying characteristics of their graphs.

Graph Behavior

The graph behavior, especially in cubic functions, is largely determined by the leading coefficient of the highest degree term. In the function \( f(x) = -x^3 + 4x \), the highest degree term is \(-x^3\).

• A positive leading coefficient in a cubic function means the graph will 'fall' to the left and 'rise' to the right.
• A negative leading coefficient, like in \(-x^3\), results in the graph 'rising' to the left and 'falling' to the right.

For the function \( f(x) = -x^3 + 4x \), the negative coefficient of the \( x^3 \) term indicates the graph behaves contrary to what is stated in the original exercise. Understanding the correct behavior allows us to make accurate predictions about the graph's shape and direction.

End Behavior of Functions

The end behavior of a function describes how the graph behaves as \( x \) approaches positive or negative infinity. This aspect is crucial for understanding the overall trend of the function beyond the immediate vicinity of roots or critical points.
For a cubic function such as \( f(x) = -x^3 + 4x \):

• As \( x \to -\infty \), the graph will rise because the leading term \(-x^3\) dominates and due to its negative coefficient, the output becomes positive.
• As \( x \to \infty \), the graph will fall for the same reason—the negative coefficient of the leading term causes the function values to become negative.

This understanding of end behavior gives you a clear idea of how the function stretches out at the extremes.

Function Symmetry

Function symmetry is a critical concept, especially when considering the type of symmetry it possesses. Odd functions like \( f(x) = -x^3 + 4x \) exhibit symmetry about the origin, which differs from even functions that show symmetry about the y-axis.

• Origin symmetry: If the point \((a, b)\) is on the graph of the function, then \((-a, -b)\) is also on the graph.
• In context: For the given function, knowing it has origin symmetry helps to understand its overall shape and trajectory.

By recognizing symmetry, especially with odd functions, you simplify both the analysis and the graphing process of the function, saving time and increasing accuracy.