To determine whether the relationship presented in the table is linear or nonlinear, and whether it represents a function, we can analyze the values of \( Y \) corresponding to the values of \( X \).
The \( Y \) values given for \( X = 1, 2, 3, 4, 5, 6 \) are:
- \( Y(1) = 1 = 1^3 \)
- \( Y(2) = 8 = 2^3 \)
- \( Y(3) = 27 = 3^3 \)
- \( Y(4) = 64 = 4^3 \)
- \( Y(5) = 125 = 5^3 \)
- \( Y(6) = 216 = 6^3 \)
From this pattern, we can see that \( Y \) equals \( X^3 \) for each corresponding \( X \).
Determination of Linearity:
- A relationship is linear if the change in \( Y \) for a consistent change in \( X \) results in a constant rate of change (slope).
- Since \( Y = X^3 \) is a cubic function, it produces a nonlinear relationship. Therefore, the table is nonlinear.
Determination of Functionality:
- A relation is a function if each input \( X \) corresponds to exactly one output \( Y \).
- In the given table, each value of \( X \) has a unique value of \( Y \). Thus, this relationship satisfies the definition of a function.
Conclusion:
- The relationship is nonlinear and it is a function.
If you continue the pattern for \( X = 7 \):
- \( Y(7) = 7^3 = 343 \).
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