Question

Let \(E\) be an even function and O be an odd function. Determine the symmetry, if any, of the following functions. $$\boldsymbol{O} \circ \boldsymbol{E}$$

Short answer

Question: Determine the symmetry of the composite function O(E(x)), if any, where E(x) is an even function and O(x) is an odd function. Answer: The composite function O(E(x)) is an even function and has symmetry with respect to the y-axis.

Step-by-step solution

Recall Definitions of Even and Odd Functions

Before starting the main calculations, remember the definitions of an even and an odd function: - Even Function: A function is "even" if the following condition holds true for all x in the domain: f(x) = f(-x) - Odd Function: A function is "odd" if the following condition holds true for all x in the domain: f(x) = -f(-x) Our goal is to determine if the composite function O(E(x)) is even, odd, or neither.

Check if O(E(x)) is an Even Function

Now, let's check if O(E(x)) is an even function by verifying if O(E(x)) = O(E(-x)): - We know that E(x) is an even function, so E(x) = E(-x) - Therefore, O(E(x)) = O(E(-x)) Since the condition of an even function is satisfied, O(E(x)) is an even function.

Conclusion

After checking, we've found that the composite function O(E(x)) is an even function. So, the composite function O(E(x)) has symmetry with respect to the y-axis.

Key concepts

Even Functions

A function is considered even when its graph exhibits a certain type of symmetry—specifically, symmetry with respect to the y-axis. This means that if you were to fold the graph along the y-axis, both halves would match perfectly.
This occurs when for every x in the domain, the equation \( f(x) = f(-x) \) holds true. Classic examples include functions like \( f(x) = x^2 \), where \( f(x) \) and \( f(-x) \) yield the same result. This symmetry allows you to predict the value of the function for a negative input by using its positive counterpart.
Even functions can be straightforward to identify because they often include only even powers of x. Remember, recognizing this symmetry can simplify understanding the function's behavior across different inputs.

Odd Functions

Odd functions showcase a different type of symmetry known as origin symmetry. For a function to be classified as odd, it must satisfy the condition \( f(x) = -f(-x) \) for all x in its domain. This means that if the function goes through the origin, it will have rotational symmetry of 180 degrees around it.
Visualizing odd functions can be fascinating: if you rotate the graph 180 degrees around the origin, it looks the same. For example, \( f(x) = x^3 \) is an odd function since flipping both the input and output signs yields the same value as rotating the graph.
Odd functions often contain only odd powers of x. This pattern helps in predicting the function's output for positive and negative pairs of an input, simplifying calculations and understanding of the function's graphical representation.

Composite Functions

Composite functions are built by applying one function to the results of another function. If you imagine each function as a machine, the output of one becomes the input of the other. In mathematical terms, if you have functions \( f \) and \( g \), the composite function \( f(g(x)) \) is formed by taking \( x \) and first putting it into \( g \), and then putting the result into \( f \).
Understanding the symmetry of composite functions, like \( O(E(x)) \), involves checking if their combined structure meets the criteria for being even or odd. In this case, with \( E \) an even function and \( O \) an odd function, \( O(E(x)) \) showcased symmetry characteristics typical of even functions.

• Even function \( E \) ensures \( E(x) = E(-x) \).
• Combining with odd function \( O \) doesn't alter the even symmetry, leading to \( O(E(x)) = O(E(-x)) \), confirming it's even.

Assembling composite functions requires attention to the interplay between their formulae to understand the resulting function's symmetry.