Practice Exercises: Identifying Function Symmetry

This page presents a set of exercises for students to practice determining the symmetry of functions. The task is to be completed without the use of a calculator, emphasizing analytical skills.

Three functions are provided for analysis:

1. f(x) = 3
2. f(x) = x · ln(x^3 - 4)
3. f(x) = 6 / (e^x + e^-x)

Students are asked to examine each function and identify any symmetries present. This exercise reinforces the concepts of axial symmetry to the y-axis and point symmetry to the origin introduced in the previous section.

Highlight: These exercises challenge students to apply their understanding of symmetry to various function types, including constant, logarithmic, and exponential functions.

Solutions: Analyzing Function Symmetry

This page provides detailed solutions to the practice exercises, demonstrating the step-by-step process of determining function symmetry.

For f(x) = 3: The solution shows that this constant function is both axially symmetric to the y-axis and point-symmetric to the origin.

Example: f(x) = f(-x) and -f(x) = f(-x) both hold true for all x, confirming both types of symmetry.

For f(x) = x · ln(x^3 - 4): The solution demonstrates that this function has neither axial nor point symmetry.

Highlight: The equations f(x) = f(-x) and -f(x) = f(-x) are both shown to be false, indicating no symmetry.

For f(x) = 6 / (e^x + e^-x): The solution reveals that this function is point-symmetric to the origin but not axially symmetric to the y-axis.

Example: The equation -f(x) = f(-x) is proven true, confirming point symmetry, while f(x) = f(-x) is shown to be false, ruling out axial symmetry.

Vocabulary: The use of hyperbolic functions (e^x + e^-x) in this example introduces students to more advanced function types and their symmetry properties.

Analysis: Symmetry in Functions

This page introduces the concept of symmetry in mathematical functions. It emphasizes two main types of symmetry: axial symmetry to the y-axis and point symmetry to the origin.

Definition: Axial symmetry to the y-axis occurs when f(x) = f(-x), while point symmetry to the origin is present when -f(x) = f(-x).

The page provides two detailed examples to illustrate these concepts:

Example 1: f(x) = 4x^5 - 8x^3 This function is tested for both types of symmetry.

Example: When checking for axial symmetry, the equation 4x^5 - 8x^3 = 4(-x)^5 - 8(-x)^3 is evaluated, revealing no axial symmetry.

Highlight: However, when testing for point symmetry, the equation -(4x^5 - 8x^3) = 4(-x)^5 - 8(-x)^3 holds true, indicating that the graph of this function is point-symmetric to the origin.

Example 2: f(x) = x^2 · ln(x^4 - 2) This function is tested for axial symmetry to the y-axis.

Example: The equation x^2 · ln(x^4 - 2) = (-x)^2 · ln((-x)^4 - 2) is shown to be true, demonstrating that the graph of this function is axially symmetric to the y-axis.

Highlight: The page notes that once axial symmetry is confirmed, there's no need to check for point symmetry.