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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}



Answer :

GinnyAnswer
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]

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