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Chapter 1: Problem 15

Determine whether the equation represents \(y\) as a function of \(x .\) $$y=\sqrt{16-x^{2}}$$

Short Answer

Expert verified
Yes, the given equation represents y as a function of x.

Step by step solution

01

Understanding the equation

Given equation is \(y=\sqrt{16-x^{2}}\). This is the equation of a semi-circle with radius 4 and centered at the origin.
02

Visual analysis

By visualizing the graph, it can be seen that for each possible x-value, there is only one corresponding y-value. Therefore, this suggests that the equation represents y as a function of x.
03

Algebraic analysis

An algebraic way to verify this is by rewriting the initial equation to \(x=\sqrt{16-y^{2}}\). By doing this, one can see that for each y-value, there are two possible x-values, which confirms that y can be represented as a function of x.
04

Conclusion

Based on the visual and algebraic analysis, it is clear that the given equation does demonstrates that y can be represented as a function of x.

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