Question

Challenge Problem If \(f\) is an odd function and \(g\) is an even function, show that the composite functions \(f \circ g\) and \(g \circ f\) are both even.

Short answer

Both f(g(x)) and g(f(x)) are even functions.

Step-by-step solution

Understand Definitions

The problem involves odd and even functions. Recall that a function f(x) is odd if f(-x) = -f(x) for all x in its domain, and a function g(x) is even if g(-x) = g(x) for all x in its domain.

Define Composite Functions

To show that the composite functions f(g(x)) and g(f(x)) are even, first write down their definitions: - Composite function 1: h(x) = f(g(x)) - Composite function 2: k(x) = g(f(x))

Show f(g(x)) is Even

Evaluate h(-x): h(-x) = f(g(-x)) Since g is even, g(-x) = g(x), so: h(-x) = f(g(x)) By the definition of h(x), we have: h(-x) = h(x) Thus, f(g(x)) is an even function.

Show g(f(x)) is Even

Evaluate k(-x): k(-x) = g(f(-x)) Since f is odd, f(-x) = -f(x), so: k(-x) = g(-f(x)) Since g is even, g(-f(x)) = g(f(x)), so: k(-x) = g(f(x)) By the definition of k(x), we have: k(-x) = k(x) Thus, g(f(x)) is an even function.

Key concepts

Odd Function

An odd function has a special symmetry around the origin. The key property of an odd function is that flipping the input (changing x to -x) will change the sign of the output. Mathematically, this is stated as: \(f(-x) = -f(x)\). This means that if you take the input x, get the output f(x), and then change x to -x, the new output will be -f(x).For example, the function \ f(x) = x^3 \ is an odd function because \ f(-x) = (-x)^3 = -x^3 = -f(x) \. Odd functions 'mirror' each other across the origin.

Even Function

Even functions are symmetrical about the y-axis. This means that if you flip the input (change x to -x), the output stays the same. Mathematically, even functions satisfy the property: \(g(-x) = g(x)\). A typical example of an even function is \ g(x) = x^2 \ because \ g(-x) = (-x)^2 = x^2 = g(x) \. Even functions are like reflections in a mirror along the y-axis.

Function Composition

Function composition is the process of applying one function to the results of another function. If you have two functions f and g, the composite function \ f \circ g \ (read as 'f of g') is defined as: \[ (f \circ g)(x) = f(g(x)) \]. This means you first apply g to x, then apply f to the result of g(x).Similarly, the composite function \ g \circ f \ is defined as: \[ (g \circ f)(x) = g(f(x)) \]. These are not necessarily the same function, as the order in which you apply them matters.

Algebra

Algebra involves manipulating mathematical symbols according to specified rules. When working with functions, algebraic operations often include addition, subtraction, multiplication, and composition. In the context of function composition, algebra helps to show how properties of individual functions (like being odd or even) can combine to yield new properties for the composite functions.By using algebraic manipulation, one can prove statements such as 'the composite of an odd function and an even function is even.' This involves substituting and simplifying expressions according to the definitions and properties of odd and even functions, as seen in the problem solution.