Question

example 1 end behavior and degree of a monomial function
describe the end behavior of ( f(x) = -2x^3 ) using the leading coefficient and degree, and state the domain and range.
the leading coefficient of ( f(x) ) is ______, which is negative.
the degree is ______, which is odd.
because the leading coefficient is negative and the degree is odd, as ( x \to -infty ), ( f(x) \to ) ____ and as ( x \to infty ), ( f(x) \to ) ____.
because this is a monomial function, the domain is all real numbers.
because the leading coefficient is negative and the degree is odd, the range is ______________.

Explanation:

Step1: Identify Leading Coefficient

The leading coefficient of \( f(x) = -2x^3 \) is the coefficient of the term with the highest power of \( x \), which is \(-2\).

Step2: Identify Degree

The degree of a monomial is the exponent of \( x \), so for \( -2x^3 \), the degree is \( 3 \).

Step3: Analyze End Behavior

For a polynomial with leading coefficient \( a \) and degree \( n \):

• If \( n \) is odd and \( a < 0 \), as \( x \to -\infty \), \( f(x) \to \infty \) (because \( (-2)(-\infty)^3 = (-2)(-\infty) = \infty \)) and as \( x \to \infty \), \( f(x) \to -\infty \) (because \( (-2)(\infty)^3 = (-2)(\infty) = -\infty \)).

Step4: Determine Domain and Range

• Domain of a monomial (polynomial) is all real numbers (\( \mathbb{R} \)) since there are no restrictions on \( x \).
• For an odd - degree polynomial with a non - zero leading coefficient, the range is also all real numbers. This is because as \( x \) takes on all real values, the function values will also cover all real numbers (since it goes from \( -\infty \) to \( \infty \) as \( x \) varies over \( \mathbb{R} \)).

Answer:

• The leading coefficient of \( f(x) \) is \(\boldsymbol{-2}\), which is negative.
• The degree is \(\boldsymbol{3}\), which is odd.
• As \( x \to -\infty \), \( f(x) \to \boldsymbol{\infty} \) and as \( x \to \infty \), \( f(x) \to \boldsymbol{-\infty} \).
• The domain is all real numbers (\( \mathbb{R} \)) and the range is all real numbers (\( \mathbb{R} \)).