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


𝔂 = e⁻ˣ²

Verified step by step guidance

1. To determine symmetry about the y-axis, check if the function y = f(x) satisfies f(x) = f(-x). Substitute -x into the function: y = e^(-(-x)^2). Simplify to see if it equals the original function.
2. To determine symmetry about the origin, check if the function y = f(x) satisfies f(-x) = -f(x). Substitute -x into the function: y = e^(-(-x)^2) and compare it to -e^(-x^2).
3. Simplify the expression e^(-(-x)^2) to e^(-x^2) and compare it to the original function e^(-x^2) to check for y-axis symmetry.
4. Compare e^(-x^2) with -e^(-x^2) to check for origin symmetry. If they are not equal, the function is not symmetric about the origin.
5. Conclude whether the function is symmetric about the y-axis, the origin, or neither based on the comparisons made in the previous steps.

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 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 equation. 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.

Exponential functions

Exponential functions, such as y = e^(-x²), are characterized by a constant base raised to a variable exponent. These functions typically exhibit rapid growth or decay. Understanding their general shape and behavior is crucial for analyzing their symmetry properties, as they often do not possess symmetry about the y-axis or the origin.