- Identify the leading term of the polynomial as \(x^5\), which dictates the end behavior.
- Calculate the limit of the leading term as \(x \to -\infty\), finding that the expression approaches \(-\infty\).
- Calculate the limit of the leading term as \(x \to +\infty\), finding that the expression approaches \(+\infty\).
- Conclude that as \(x\) goes to negative infinity, \(f(x)\) goes to \(\boxed{-\infty}\) and as \(x\) goes to positive infinity, \(f(x)\) goes to \(\boxed{+\infty}\).
Explanation
- Analyzing the Polynomial Function
The function is given by \(f(x) = x^5 - 9x^3\). This is a polynomial function of degree 5, which is an odd number. The leading coefficient is 1, which is positive. The end behavior of a polynomial is determined by its leading term, \(x^5\), as \(x\) grows large in magnitude.
- Determining End Behavior as \(x \to -\infty\)
To find the limit as \(x \to -\infty\), we examine the behavior of the leading term: \(\lim_{x \to -\infty} (x^5 - 9x^3) = \lim_{x \to -\infty} x^5(1 - \frac{9}{x^2}) = (-\infty)^5(1 - 0) = -\infty\). Since the power is odd and the coefficient is positive, the function decreases toward negative infinity as \(x\) decreases.
- Determining End Behavior as \(x \to +\infty\)
To find the limit as \(x \to +\infty\), we examine the behavior of the leading term: \(\lim_{x \to +\infty} (x^5 - 9x^3) = \lim_{x \to +\infty} x^5(1 - \frac{9}{x^2}) = (+\infty)^5(1 - 0) = +\infty\). Since the power is odd and the coefficient is positive, the function increases toward positive infinity as \(x\) increases.
- Final Conclusion
Based on the limits evaluated, as \(x \to -\infty\), \(f(x) \to -\infty\), and as \(x \to +\infty\), \(f(x) \to +\infty\). Therefore, the completed statements are: 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. The final results are \(\boxed{-\infty}\) and \(\boxed{+\infty}\).
Examples
Understanding the end behavior of functions is fundamental in fields like economics or physics, where you might model long-term trends. For example, if a company's profit is modeled by a polynomial \(P(t) = at^n + \dots\), knowing the leading term's behavior helps predict whether the long-term profit will grow boundlessly or fall into massive debt as time \(t\) increases. Analyzing end behavior allows planners to prepare for extreme outcomes.