To analyze the end behavior of the function \( j(x) = \frac{12}{5}x^3 - \frac{27}{5}x \), we focus on the leading term, since it will dominate the behavior of the function as \( x \) approaches positive or negative infinity.
The leading term of the function is \( \frac{12}{5}x^3 \). We consider the following:
- Degree of the polynomial: The polynomial \( j(x) \) is of degree 3 (since the highest exponent of \( x \) is 3).
- Leading coefficient: The leading coefficient is \( \frac{12}{5} \), which is positive.
End Behavior Analysis:
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As \( x \to \infty \) (positive infinity):
- The term \( \frac{12}{5}x^3 \) dominates. Since it is a cubic term with a positive coefficient, \( j(x) \) will approach \( +\infty \).
- Therefore, \( j(x) \to +\infty \) as \( x \to +\infty \).
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As \( x \to -\infty \) (negative infinity):
- Again, the leading term \( \frac{12}{5}x^3 \) dominates. In this case, as \( x \) becomes more negative, \( x^3 \) also becomes more negative, thus \( j(x) \) will approach \( -\infty \).
- Therefore, \( j(x) \to -\infty \) as \( x \to -\infty \).
Conclusion:
In summary, the end behavior of the function \( j(x) \) can be stated as follows:
- As \( x \to +\infty \), \( j(x) \to +\infty \).
- As \( x \to -\infty \), \( j(x) \to -\infty \).
This behavior is typical for an odd-degree polynomial with a positive leading coefficient.