In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.


𝔂 = x²/⁵

Verified step by step guidance

1. To determine symmetry about the y-axis, check if replacing x with -x in the function yields the same function. For the given function y = x^(2/5), replace x with -x to get y = (-x)^(2/5).
2. Simplify the expression (-x)^(2/5). Since the exponent 2/5 is a positive rational number, (-x)^(2/5) simplifies to x^(2/5) because the square of a negative number is positive.
3. Since y = (-x)^(2/5) simplifies to y = x^(2/5), the function is symmetric about the y-axis.
4. To check for symmetry about the origin, replace both x with -x and y with -y in the original equation. This gives -y = (-x)^(2/5).
5. Since -y = x^(2/5) does not simplify to the original equation y = x^(2/5), the function is not symmetric about the origin. Therefore, the graph of the function is symmetric about the y-axis only.

Key Concepts

Symmetry about the y-axis

A function is symmetric about the y-axis if replacing x with -x in the function yields the same output. Mathematically, this means that f(-x) = f(x) for all x in the domain of the function. This type of symmetry indicates that the graph of the function is a mirror image across the y-axis.

Symmetry about the origin

A function is symmetric about the origin if replacing x with -x and y with -y results in the same function. This is expressed as f(-x) = -f(x). Functions with this symmetry exhibit rotational symmetry of 180 degrees around the origin, meaning that if you rotate the graph, it looks the same.

Analyzing function symmetry

To determine the symmetry of a function, one can evaluate the function at both x and -x. By comparing the results, one can conclude whether the function is symmetric about the y-axis, the origin, or neither. This analysis is crucial for understanding the behavior of the graph and its visual representation.