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Chapter 3: Problem 20

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=11 x^{3}-6 x^{2}+x+3$$

Short Answer

Expert verified
The end behavior of the function is: as x approaches positive infinity, f(x) approaches positive infinity and as x approaches negative infinity, f(x) approaches negative infinity.

Step by step solution

01

Recognize the Leading Term

In the function \(f(x) = 11x^3 - 6x^2 + x + 3\), the term with the highest degree is 11x^3. This is the leading term.
02

Apply the Leading Coefficient Test

The Leading Coefficient Test states that if the degree of the polynomial is odd and the leading coefficient is positive, then as x approaches positive infinity, f(x) approaches positive infinity and as x approaches negative infinity, f(x) approaches negative infinity.
03

Determine End Behavior

The degree of the leading term in the function is 3 which is odd and the leading coefficient is 11 which is positive. Hence, as x approaches positive infinity, f(x) approaches positive infinity, and as x approaches negative infinity, f(x) approaches negative infinity.

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