To analyze the characteristics of the parent linear function, which is typically represented as \( f(x) = x \), let's evaluate each statement:
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The domain and range are \( (-\infty, \infty) \):
- This is true. The parent linear function can accept any real number as input (domain) and can produce any real number as output (range).
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It is an even function:
- This is false. A function is considered even if \( f(-x) = f(x) \) for all \( x \). The parent linear function \( f(x) = x \) does not satisfy this property because \( f(-x) = -x \), which is not equal to \( f(x) \).
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It decreases over the interval \( (-\infty, \infty) \):
- This is false. The parent linear function \( f(x) = x \) is neither decreasing nor increasing across its entire domain; it is a straight line with a slope of 1, meaning it increases.
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The end behavior is \( f(x) \to \infty \) as \( x \to \infty \) and \( f(x) \to -\infty \) as \( x \to -\infty \):
- This is true. As \( x \) increases without bound, \( f(x) = x \) goes to positive infinity, and as \( x \) decreases without bound, \( f(x) = x \) goes to negative infinity.
Based on this analysis, the correct characteristics of the graph of the parent linear function are:
- The domain and range are \( (-\infty, \infty) \).
- The end behavior is \( f(x) \to \infty \) as \( x \to \infty \) and \( f(x) \to -\infty \) as \( x \to -\infty \).