Question

Challenge Problem If \(f\) and \(g\) are odd functions, show that the composite function \(f^{\circ} \mathrm{g}\) is also odd.

Short answer

The composite function \( f \circ g \) is odd because for odd functions \(f\) and \(g\), \( (f \circ g)(-x) = -(f \circ g)(x) \).

Step-by-step solution

Understand the Definition of Odd Functions

Recall that a function is odd if for every real number x, the function satisfies the condition: \[ f(-x) = -f(x) \] This means that the function is symmetric with respect to the origin.

Define the Composite Function

The composite function \(f \circ g\) of two functions \(f\) and \(g\) is defined as: \[ (f \circ g)(x) = f(g(x)) \] Here, we need to show that \(f \circ g\) is odd if both \(f\) and \(g\) are odd.

Apply the Odd Functions Property to the Composite Function

To determine if \(f \circ g\) is odd, we need to show that: \[ (f \circ g)(-x) = - (f \circ g)(x) \] Start by evaluating \((f \circ g)(-x)\).

Evaluate \( (f \circ g)(-x) \)

Using the definition of the composite function: \[ (f \circ g)(-x) = f(g(-x)) \] Now, use the fact that \(g\) is an odd function.

Apply the Odd Property of \(g\)

Since \(g\) is odd: \[ g(-x) = -g(x) \] Substitute \(-g(x)\) back into the composite function: \[ (f \circ g)(-x) = f(-g(x)) \]

Apply the Odd Property of \(f\)

Since \(f\) is odd: \[ f(-g(x)) = -f(g(x)) \] Therefore: \[ (f \circ g)(-x) = -f(g(x)) \]

Conclude that the Composite Function is Odd

We can see from our evaluation that: \[ (f \circ g)(-x) = -f(g(x)) = -(f \circ g)(x) \] Thus, we have shown that the composite function \( f \circ g \) is also odd.

Key concepts

Composite Functions

A composite function is formed when one function is applied to the result of another function. This can be written as \( (f \circ g)(x) = f(g(x)) \). Essentially, you first apply function g to x, and then apply function f to the result. Composite functions allow us to combine multiple transformations in a sequence. It's like a process where one operation feeds into another. Understanding how composite functions work is crucial in various fields of mathematics and applied sciences.

Function Symmetry: Odd Functions

Odd functions have a specific type of symmetry known as origin symmetry. This means that a function f is odd if \( f(-x) = -f(x) \) for all x in the function's domain. When graphed, odd functions exhibit symmetry around the origin. If you reflect the function across both the x-axis and y-axis, you will end up with the same function. Common examples of odd functions include \( f(x) = x^3 \) and \( f(x) = \sin(x) \). Recognizing this symmetry helps in solving problems and simplifying function analysis.

Function Properties: Composite of Odd Functions

When combining two odd functions in a composite function, we can conclude that the resulting function is also odd. Here's a quick look at the proof:
(1) Start with odd functions f and g such that \( f(-x) = -f(x) \) and \( g(-x) = -g(x) \).
(2) The composite function is defined as \( (f \circ g)(x) = f(g(x)) \).
(3) To show that \( (f \circ g)(x) \) is odd, evaluate \( (f \circ g)(-x) \):
(4) Using the definition of \( f \circ g \), we get \( (f \circ g)(-x) = f(g(-x)) \).
(5) Since g is odd, \( g(-x) = -g(x) \). Substitute this back:
(6) \ f(-g(x)) \.
(7) Since f is odd, \( f(-g(x)) = -f(g(x)) \).
(8) Hence, \( (f \circ g)(-x) = -f(g(x)) = -(f \circ g)(x) \). This demonstrates that the composite function \( f \circ g \) is also odd. This property is useful as it helps predict the nature of complex function combinations.