Answered

IDNLearn.com offers expert insights and community wisdom to answer your queries. Get step-by-step guidance for all your technical questions from our knowledgeable community members.

Problem (1). A function [tex]\( f \)[/tex] is called even if [tex]\( f(-x)=f(x) \)[/tex] for all [tex]\( x \)[/tex]. For example, [tex]\( f(x)=x^2 \)[/tex] or [tex]\( \cos(x) \)[/tex] are even functions. A function is called odd if [tex]\( f(-x)=-f(x) \)[/tex] for all [tex]\( x \)[/tex]. For example, [tex]\( \sin(x) \)[/tex], [tex]\( \tan(x) \)[/tex], or [tex]\( f(x)=x^3 \)[/tex] are odd functions.

Suppose [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are two functions. Classify if the functions [tex]\( h(x)=f(x) g(x) \)[/tex] and [tex]\( i(x)=f(g(x)) \)[/tex] are even or odd based on whether [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are even or odd. Fill in the table below with your answer.

\begin{tabular}{c|c|c|c}
[tex]\( f \)[/tex] & [tex]\( g \)[/tex] & [tex]\( h \)[/tex] & [tex]\( i \)[/tex] \\
\hline
even & even & even & even \\
even & odd & odd & even \\
odd & even & odd & odd \\
odd & odd & even & odd \\
\end{tabular}

Sagot :

GinnyAnsweridn
Certainly! Let's analyze the functions [tex]\( h(x) = f(x)g(x) \)[/tex] and [tex]\( i(x) = f(g(x)) \)[/tex] based on the classifications of [tex]\( f \)[/tex] and [tex]\( g \)[/tex] as either even or odd.

### Analysis of [tex]\( h(x) = f(x) g(x) \)[/tex]
#### Case 1: [tex]\( f(x) \)[/tex] is even and [tex]\( g(x) \)[/tex] is even
- For an even function [tex]\( f \)[/tex], we have [tex]\( f(-x) = f(x) \)[/tex].
- For an even function [tex]\( g \)[/tex], we have [tex]\( g(-x) = g(x) \)[/tex].
- Therefore, [tex]\( h(-x) = f(-x)g(-x) = f(x)g(x) = h(x) \)[/tex].
- Conclusion: [tex]\( h(x) \)[/tex] is even.

#### Case 2: [tex]\( f(x) \)[/tex] is even and [tex]\( g(x) \)[/tex] is odd
- For an even function [tex]\( f \)[/tex], we have [tex]\( f(-x) = f(x) \)[/tex].
- For an odd function [tex]\( g \)[/tex], we have [tex]\( g(-x) = -g(x) \)[/tex].
- Therefore, [tex]\( h(-x) = f(-x)g(-x) = f(x)(-g(x)) = -f(x)g(x) = -h(x) \)[/tex].
- Conclusion: [tex]\( h(x) \)[/tex] is odd.

#### Case 3: [tex]\( f(x) \)[/tex] is odd and [tex]\( g(x) \)[/tex] is even
- For an odd function [tex]\( f \)[/tex], we have [tex]\( f(-x) = -f(x) \)[/tex].
- For an even function [tex]\( g \)[/tex], we have [tex]\( g(-x) = g(x) \)[/tex].
- Therefore, [tex]\( h(-x) = f(-x)g(-x) = (-f(x))g(x) = -f(x)g(x) = -h(x) \)[/tex].
- Conclusion: [tex]\( h(x) \)[/tex] is odd.

#### Case 4: [tex]\( f(x) \)[/tex] is odd and [tex]\( g(x) \)[/tex] is odd
- For an odd function [tex]\( f \)[/tex], we have [tex]\( f(-x) = -f(x) \)[/tex].
- For an odd function [tex]\( g \)[/tex], we have [tex]\( g(-x) = -g(x) \)[/tex].
- Therefore, [tex]\( h(-x) = f(-x)g(-x) = (-f(x))(-g(x)) = f(x)g(x) = h(x) \)[/tex].
- Conclusion: [tex]\( h(x) \)[/tex] is even.

### Analysis of [tex]\( i(x) = f(g(x)) \)[/tex]
#### Case 1: [tex]\( f(x) \)[/tex] is even and [tex]\( g(x) \)[/tex] is even
- For an even function [tex]\( f \)[/tex], we have [tex]\( f(-x) = f(x) \)[/tex].
- For an even function [tex]\( g \)[/tex], we have [tex]\( g(-x) = g(x) \)[/tex].
- Therefore, [tex]\( i(-x) = f(g(-x)) = f(g(x)) = i(x) \)[/tex].
- Conclusion: [tex]\( i(x) \)[/tex] is even.

#### Case 2: [tex]\( f(x) \)[/tex] is even and [tex]\( g(x) \)[/tex] is odd
- For an even function [tex]\( f \)[/tex], we have [tex]\( f(-x) = f(x) \)[/tex].
- For an odd function [tex]\( g \)[/tex], we have [tex]\( g(-x) = -g(x) \)[/tex].
- Therefore, [tex]\( i(-x) = f(g(-x)) = f(-g(x)) = f(g(x)) = i(x) \)[/tex].
- Conclusion: [tex]\( i(x) \)[/tex] is even.

#### Case 3: [tex]\( f(x) \)[/tex] is odd and [tex]\( g(x) \)[/tex] is even
- For an odd function [tex]\( f \)[/tex], we have [tex]\( f(-x) = -f(x) \)[/tex].
- For an even function [tex]\( g \)[/tex], we have [tex]\( g(-x) = g(x) \)[/tex].
- Therefore, [tex]\( i(-x) = f(g(-x)) = f(g(x)) = i(x) \)[/tex].
- Conclusion: [tex]\( i(x) \)[/tex] is odd.

#### Case 4: [tex]\( f(x) \)[/tex] is odd and [tex]\( g(x) \)[/tex] is odd
- For an odd function [tex]\( f \)[/tex], we have [tex]\( f(-x) = -f(x) \)[/tex].
- For an odd function [tex]\( g \)[/tex], we have [tex]\( g(-x) = -g(x) \)[/tex].
- Therefore, [tex]\( i(-x) = f(g(-x)) = f(-g(x)) = -f(g(x)) = -i(x) \)[/tex].
- Conclusion: [tex]\( i(x) \)[/tex] is odd.

### Summary Table:
[tex]\[ \begin{array}{c|c|c|c} f & g & h & i \\ \hline \text{even} & \text{even} & \text{even} & \text{even} \\ \text{even} & \text{odd} & \text{odd} & \text{even} \\ \text{odd} & \text{even} & \text{odd} & \text{odd} \\ \text{odd} & \text{odd} & \text{even} & \text{odd} \\ \end{array} \][/tex]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Thank you for visiting IDNLearn.com. We’re here to provide dependable answers, so visit us again soon.