Understanding the Domain and Range of the Polynomial Function 2x^3 - 3x^26

Polynomial functions are a fundamental part of calculus and mathematical analysis. In this article, we will explore the domain and range of the polynomial function:

Function: ( f(x) 2x^3 - 3x^{26} )

1. Domain of the Function

The domain of a function is the set of all possible input values (x-values) that will produce a valid output. For the polynomial function ( f(x) 2x^3 - 3x^{26} ), there are no restrictions on the values of ( x ) that can be input into the function. This is because any real number can be raised to any power, and the resulting expression will always be a real number.

Conclusion: The domain of the function is all real numbers, denoted as R.

2. Range of the Function

The range of a function is the set of all possible output values (y-values) that the function can produce. To determine the range, we need to analyze the behavior of the function as ( x ) approaches positive and negative infinity.

Step 1: Calculate the derivative of the function

The derivative of a function helps us understand its behavior and identify any critical points. Let's find the derivative of ( f(x) 2x^3 - 3x^{26} ).

[ f'(x) frac{d}{dx}(2x^3 - 3x^{26}) 6x^2 - 3 cdot 26x^{25} 6x^2 - 78x^{25} ]

Step 2: Analyze the behavior as ( x ) approaches ±∞

As ( x ) approaches positive or negative infinity, the term ( -3x^{26} ) will dominate the function. Therefore, the function will behave similarly to ( -3x^{26} ).

[ lim_{x to infty} f(x) -infty ]

[ lim_{x to -infty} f(x) infty ]

Conclusion: As x approaches positive infinity, ( f(x) ) approaches negative infinity, and as x approaches negative infinity, ( f(x) ) approaches positive infinity. Therefore, the function can take on any real number as a y-value.

Conclusion: The range of the function is all real numbers, denoted as R.

3. Continuity and Monotonicity

A function is continuous on its domain if there are no breaks or jumps in the graph of the function. Since the function ( f(x) 2x^3 - 3x^{26} ) is a polynomial, it is continuous on its entire domain, which is all real numbers.

By examining the derivative, ( f'(x) 6x^2 - 78x^{25} ), we can determine the intervals where the function is increasing or decreasing. The critical points are where the derivative is zero or undefined.

[ 6x^2 - 78x^{25} 0 ]

Solving for ( x ), we get:

[ 6x^{2(1 - 25/2)} 0 ]

[ 6x^{2 - 25} 0 ]

[ x 0 end{math}

Conclusion: The function is increasing on the intervals where ( f'(x) > 0 ) and decreasing where ( f'(x)

4. Summary

In summary, the domain and range of the function ( f(x) 2x^3 - 3x^{26} ) are as follows:

Domain: All real numbers, denoted as R. Range: All real numbers, denoted as R.

This polynomial function is continuous on its entire domain and can take on any real value, making its range all real numbers.

Key Takeaways

1. Domain: The set of all possible input values, which for this polynomial is all real numbers.

2. Range: The set of all possible output values, which for this polynomial is all real numbers.

3. Continuity: The function is continuous on all real numbers.

4. Monotonicity: The function is increasing on all real numbers.