Question

A) Does the relation \(x = y^2\) represent \(y\) as a function of \(x\)?

a) Yes, the relation represents a function.
b) No, the relation does not represent a function.

b) Does the relation \(x^2 + y^2 = 9\) represent \(y\) as a function of \(x\)?
a) Yes, the relation represents a function.
b) No, the relation does not represent a function.

Answer 1

The relation x = y^2 does not represent y as a function of x, as it fails the vertical line test, having two possible y values for a given positive x. Similarly, x^2 + y^2 = 9 does not represent y as a function of x since it corresponds to a circle with multiple y values for most x within (-3, 3).

Step-by-step explanation:

To determine whether a given relationship represents y as a function of x, we must consider whether each input x corresponds to exactly one output y. This is also known as the vertical line test.

a) The relation x = y^2 does not represent y as a function of x because for a given positive value of x, there are two possible values for y (one positive and one negative). Hence, this relation fails the vertical line test.

b) The relation x^2 + y^2 = 9 represents a circle with a radius of 3. For most values of x within the interval (-3, 3), there are two possible values of y, corresponding to the points above and below the x-axis. Thus, this relation also fails to represent y as a function of x.

Answer 2

a) No, the relation \(x = y^2\) does not represent (y) as a function of (x).

b) Yes, the relation \(x^2 + y^2 = 9\) does not represent (y) as a function of (x).

Step-by-step explanation:

a) For a relation to represent \(y\) as a function of \(x\), each \(x\) value should correspond to only one \(y\) value. In the case of \(x = y^2\), for certain \(x\) values, there are two possible \(y\) values (positive and negative square roots), violating the vertical line test. Therefore, the relation \(x = y^2\) does not represent \(y\) as a function of \(x\).

b) The relation \(x^2 + y^2 = 9\) represents a circle centered at the origin with radius 3. For a function, each \(x\) value should correspond to only one \(y\) value. In this case, for each \(x\), there are two corresponding \(y\) values (positive and negative square roots), violating the vertical line test. Hence, the relation \(x^2 + y^2 = 9\) does not represent \(y\) as a function of \(x\).

In conclusion, understanding the concept of functions and applying the vertical line test is crucial to determine whether a relation represents \(y\) as a function of \(x\). The vertical line test helps identify situations where a single \(x\) value maps to multiple \(y\) values, indicating a non-functional relation.