To analyze the relation provided, we can look closely at the values of \( f(x) \) based on the given \( x \) values.
Given:
- \( x: -2, 0, 2, 4 \)
- \( f(x): 4, 0, 4, 16 \)
- Domain: { -2, 0, 2, 4 }
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To determine if the relation is a function: A relation is a function if each input (or \( x \) value) is related to exactly one output (or \( f(x) \) value). In this case, we observe the outputs:
- \( f(-2) = 4 \)
- \( f(0) = 0 \)
- \( f(2) = 4 \)
- \( f(4) = 16 \)
While \( f(-2) \) and \( f(2) \) both yield the output of 4, for different values of \( x \), this still means each input has a single output. Thus, this relation is indeed a function.
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Verifying specific function representations:
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\( f(x) = x^2 \): Testing this function:
- \( f(-2) = (-2)^2 = 4 \)
- \( f(0) = (0)^2 = 0 \)
- \( f(2) = (2)^2 = 4 \)
- \( f(4) = (4)^2 = 16 \) This matches the outputs given, so \( f(x) = x^2 \) is a valid representation.
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\( f(4) = -2 \): Based on the output table, \( f(4) = 16 \), so this statement is false.
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\( f(x) = -2x \): Testing this function:
- \( f(-2) = -2(-2) = 4 \)
- \( f(0) = -2(0) = 0 \)
- \( f(2) = -2(2) = -4 \) (not matching)
- \( f(4) = -2(4) = -8 \) (not matching) Hence, \( f(x) = -2x \) is not a valid representation.
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So the correct representations or descriptions that model the relation are:
- The relation is a function.
- \( f(x) = x^2 \)
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