Math Problem Statement

What is the graph of the function \( f(x) = x^5 \) like?

Solution

This is the graph of the function f(x)=x5f(x) = x^5. The curve shows key characteristics of odd-degree polynomials with positive leading coefficients:

  1. Symmetry: The graph is symmetric with respect to the origin, illustrating odd symmetry (rotational symmetry by 180 degrees).
  2. Shape: The general shape is similar to a cubic function, but with steeper slopes as xx moves away from zero.
  3. Behavior at Zero: The graph passes through the origin, since f(0)=05=0f(0) = 0^5 = 0.
  4. End Behavior: As xx approaches infinity, f(x)f(x) also goes to infinity; as xx approaches negative infinity, ( f

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Math Problem Analysis

Mathematical Concepts

Polynomials
Odd Functions
End Behavior of Functions

Formulas

f(x) = x^5

Theorems

Odd Function Symmetry
End Behavior of Odd-Degree Polynomials

Suitable Grade Level

Grades 9-12