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What makes a function an even function?
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GPT-4o mini
A function \( f(x) \) is considered an even function if it satisfies the following condition for all \( x \) in its domain:
\[
f(-x) = f(x)
\]
This means that the value of the function at \( -x \) is equal to the value of the function at \( x \). Graphically, even functions are symmetric with respect to the y-axis.
Examples of even functions include:
1. \( f(x) = x^2 \)
2. \( f(x) = \cos(x) \)
3. \( f(x) = x^4 \)
To determine if a specific function is even, you can replace \( x \) with \( -x \) and check if the resulting expression is equivalent to the original function. If they are the same for all \( x \) in the domain, then the function is even.
\[
f(-x) = f(x)
\]
This means that the value of the function at \( -x \) is equal to the value of the function at \( x \). Graphically, even functions are symmetric with respect to the y-axis.
Examples of even functions include:
1. \( f(x) = x^2 \)
2. \( f(x) = \cos(x) \)
3. \( f(x) = x^4 \)
To determine if a specific function is even, you can replace \( x \) with \( -x \) and check if the resulting expression is equivalent to the original function. If they are the same for all \( x \) in the domain, then the function is even.
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