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For the following exercises, determine whether the relation represents y as a function of
X.

For the following exercises, determine whether the relation represents y as a function-example-1

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Answer:

7) Yes, the relation represents y as a function of x.

8) Yes, the relation represents y as a function of x.

9) Yes, the relation represents y as a function of x.

Explanation:

A function is a special type of relationship where each input (x-value) is related to exactly one output (y-value). If an input maps to multiple outputs, it would not satisfy the definition of a function and would instead be a relation.

The vertical line test is a graphical method used to determine whether a given curve or graph represents a function. If any vertical line intersects the curve at more than one point, it does not qualify as a function.

Question 7

The relation y = x² represents y as a function of x.

In this case, each x-value uniquely determines a y-value, so it satisfies the definition of a function.

The graph of y = x² is a parabola that opens upwards. Any vertical line will intersect the parabola at only one point. Therefore, it passes the vertical line test, and so is a function.

Question 8

The relation 3x² - 14 = y represents y as a function of x.

In this case, each x-value uniquely determines a y-value, so it satisfies the definition of a function.

The graph of 3x² - 14 = y is a parabola that opens upwards. Any vertical line will intersect the parabola at only one point. Therefore, it passes the vertical line test, and so is a function.

Question 9

The relation y = 2/x represents y as a function of x.

In this case, each x-value uniquely determines a y-value, so it satisfies the definition of a function.

The graph of y = 2/x is the graph of a rational function, with a vertical asymptote at x = 0 and a horizontal asymptote at y =0. This means that the relation is undefined when x = 0 or y = 0. For the defined domain of the relation, any vertical line will intersect the curve at only one point. Therefore, it passes the vertical line test, and so is a function.

Summary

All three of the given relations represent y as a function of x because they satisfy the condition that each x-value is related to exactly one y-value, and each pass the vertical line test.

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