To analyze the end behavior of the function \( j(x) = \frac{12x^3}{5} - \frac{27x}{5} \), we start by identifying the leading term of the polynomial.
The given function is a polynomial of degree 3 (the highest power of \( x \) is 3). The leading term is \( \frac{12x^3}{5} \).
End Behavior Analysis:
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Independent of Lower-Degree Terms: When analyzing the end behavior of a polynomial, we focus primarily on the leading term as \( x \) approaches positive and negative infinity. The lower-degree terms, in this case \( -\frac{27x}{5} \), will have a diminishing effect on the value of \( j(x) \) compared to the leading term as \( |x| \) increases.
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Behavior as \( x \to +\infty \):
- As \( x \) approaches positive infinity, \( \frac{12x^3}{5} \) grows very large since the cubic term dominates. Therefore: \[ j(x) \to +\infty \]
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Behavior as \( x \to -\infty \):
- As \( x \) approaches negative infinity, \( \frac{12x^3}{5} \) becomes very large and negative because the cubic term of negative \( x \) will also be negative. Thus: \[ j(x) \to -\infty \]
Conclusion:
The end behavior of the function \( j(x) = \frac{12x^3}{5} - \frac{27x}{5} \) can be summarized as follows:
- As \( x \to +\infty \), \( j(x) \to +\infty \).
- As \( x \to -\infty \), \( j(x) \to -\infty \).
This behavior is consistent with the properties of cubic polynomials, where the leading coefficient (in this case, positive) determines the function's behavior in the respective directions of the \( x \)-axis.