Problem 81 - Calculus

Question

Are the statements true or false? Give an explanation for your answer. If \(g(x)\) is an even function then \(f(g(x))\) is even for every function \(f(x)\).

Step-by-Step Solution

Answer

False; counterexamples show that \(f(g(x))\) is not necessarily even for all \(f(x)\).

Step 1: Understand the Given Statement

We are asked to determine the truth value of the statement: "If \(g(x)\) is an even function, then \(f(g(x))\) is even for every function \(f(x)\)." We know that an even function satisfies the condition \(g(-x) = g(x)\) for all \(x\).

Step 2: Analyze the Implication for Function Composition

Because \(g(x)\) is even, it results in \(g(-x) = g(x)\). Substituting these into the composition \(f(g(x))\), we have \(f(g(-x)) = f(g(x))\) for \(f(g(x))\) to be even. We must determine if this holds true for every function \(f(x)\).

Step 3: Examine Edge Cases

Consider an example where \(f(x) = x\), a linear (odd) function. Substituting this into \(f(g(x))\), we get \(f(g(x)) = g(x)\). Consequently, \(f(g(-x)) = g(-x) = g(x)\), and thus \(f(g(x)) = g(x)\). Here, \(f(g(x))\) is even because \(g(x)\) itself is even.

Step 4: Test a Counterexample

Now, consider \(f(x) = -x\). For this function, \(f(g(x)) = -g(x)\). Then \(f(g(-x)) = -g(-x) = -g(x)\) by the even property of \(g(x)\). Now, \(-g(-x) = -g(x)\), so \(-g(x) = g(x)\) implies \(f(g(x)) = -g(x)\), which is not an even function since \(f(g(-x)) = -g(x)\) does not equal \(-g(x)\).

Step 5: Conclude the Analysis

From the examination of \(f(x) = -x\), we see that there exists a function \(f(x)\) for which \(f(g(x))\) is not even despite \(g(x)\) being even. Thus, the original statement does not hold true for every function \(f(x)\).

Key Concepts

Function Composition

Function composition is a way of combining two functions where the output of one function becomes the input of another. This is typically written as \(f(g(x))\), which means you first apply \(g\) to \(x\) and then apply \(f\) to the result of \(g(x)\).

When composing functions, it's essential to pay attention to their properties because this combination can affect the outcome significantly. For example, if you start with an even function, the result after composing it with another function depends on that second function's properties.

This process sometimes leads to unexpected outcomes, showing the importance of thoroughly understanding each function's characteristics before composing them.

Odd Function

An odd function is characterized by its symmetry through the origin, meaning it satisfies the condition \(f(-x) = -f(x)\) for all \(x\).

This property contrasts with even functions, which reflect symmetry across the y-axis. Odd functions have distinct behavior that can change the nature of a composition.

For example:

• If \(f(x) = x\), then \(f(-x) = -x\), indicating it's odd.
• When composing an even function with an odd function, such as \(f(x) = -x\) and \(g(x) = x^2\), the composed function \(f(g(x)) = -x^2\) becomes neither even nor odd, as seen in the counterexample analysis.

Understanding these properties helps predict the outcome when different types of functions are composed.

Counterexample Analysis

Counterexample analysis involves finding a specific example that disproves a general statement. It's a powerful method for showing that a hypothesis or claim is not universally true.

In the case of determining whether \(f(g(x))\) is even for any \(f(x)\), despite \(g(x)\) being even, we looked at various functions to test this.

One counterexample used \(f(x) = -x\). Although \(g(x)\) remains even, \(f(g(x)) = -g(x)\) does not satisfy the criteria of being even, as \(f(g(-x))\) does not equal \(f(g(x))\).

Through this analysis, the initial claim is proven false, thus highlighting the importance of counterexamples in mathematical reasoning and analysis.

Function Properties

Function properties include various characteristics like being even or odd, being linear, or satisfying specific symmetry. These properties directly influence function outcomes, especially in composition.

Even functions have symmetric y-axis reflections, expressed as \(g(-x) = g(x)\). Odd functions possess origin symmetry, meaning \(f(-x) = -f(x)\).

Understanding these helps predict outcomes when you substitute one function into another. Here are some critical attributes to consider:

• Evenness: Function outputs mirror across the y-axis.
• Oddness: Outputs reflect symmetry through the origin.
• Linearity: Simple algebraic expressions can alter symmetry perceptions.

These properties help to comprehend how functions behave and interact, ultimately aiding in the analysis and composition of functions, as in the exercise challenge presented.