Complete the statements about the key features of the graph of [tex]f(x)=x^5-9 x^3[/tex].
As [tex]x[/tex] goes to negative infinity, [tex]f(x)[/tex] goes to $\square$ infinity, and as [tex]x[/tex] goes to positive infinity, [tex]f(x)[/tex] goes to $\square$ infinity.
Answer (2)
As x goes to negative infinity, f ( x ) approaches − ∞ , and as x goes to positive infinity, f ( x ) approaches + ∞ .
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The function is f ( x ) = x 5 − 9 x 3 .
The end behavior is determined by the leading term, x 5 .
As x approaches negative infinity, f ( x ) approaches negative infinity.
As x approaches positive infinity, f ( x ) approaches positive infinity. Therefore, the answer is: As x goes to negative infinity, f ( x ) goes to − ∞ infinity, and as x goes to positive infinity, f ( x ) goes to ∞ infinity.
Explanation
Understanding the Problem We are asked to describe the end behavior of the function f ( x ) = x 5 − 9 x 3 . This means we need to determine what happens to the value of f ( x ) as x approaches positive infinity and negative infinity.
Identifying the Leading Term The function f ( x ) is a polynomial. The end behavior of a polynomial is determined by its leading term, which is the term with the highest power of x . In this case, the leading term is x 5 .
Analyzing Behavior as x Approaches Positive Infinity As x approaches positive infinity (i.e., x becomes very large and positive), x 5 also becomes very large and positive. Therefore, as x goes to positive infinity, f ( x ) also goes to positive infinity.
Analyzing Behavior as x Approaches Negative Infinity As x approaches negative infinity (i.e., x becomes very large and negative), x 5 also becomes very large and negative. This is because a negative number raised to an odd power is negative. Therefore, as x goes to negative infinity, f ( x ) also goes to negative infinity.
Final Answer In summary, as x goes to negative infinity, f ( x ) goes to negative infinity, and as x goes to positive infinity, f ( x ) goes to positive infinity.
Examples
Understanding the end behavior of functions is crucial in many real-world applications. For example, when modeling population growth, the end behavior of the function tells us whether the population will continue to grow indefinitely or stabilize at a certain level. Similarly, in physics, the end behavior of a function can describe the long-term behavior of a system, such as the trajectory of a projectile or the decay of a radioactive substance. By analyzing the end behavior, we can make predictions about the future and gain insights into the underlying processes.
