Which equation is a function of $x$?

$x=5$

$x-y^2=9$

$x^2=y$

$x^2=y^2+16$

Analyze each equation to see if y can be uniquely determined by x .
x = 5 is not a function of x because it doesn't express y in terms of x .
x − y 2 + 9 = 0 gives y = ± x + 9 ​ , which is not a function because of the ± .
x 2 = y gives y = x 2 , which is a function of x .
x 2 = y 2 + 16 gives y = ± x 2 − 16 ​ , which is not a function because of the ± .
The equation that represents y as a function of x is x 2 = y ​ .

Explanation

Analyzing the Problem We are asked to identify which of the given equations represents y as a function of x . In other words, for which equation can we express y uniquely in terms of x ? Let's analyze each equation.

Analyzing Equation 1

x = 5 : This equation represents a vertical line where x is always 5, regardless of the value of y . It does not express y as a function of x .

Analyzing Equation 2

x − y 2 + 9 = 0 : We can rearrange this equation to solve for y :


y 2 = x + 9
y = ± x + 9 ​
For a single value of x , there are two possible values of y (a positive and a negative square root), so this is not a function of x .

Analyzing Equation 3

x 2 = y : This equation directly expresses y in terms of x : y = x 2 . For each value of x , there is only one corresponding value of y . Therefore, this is a function of x .

Analyzing Equation 4

x 2 = y 2 + 16 : We can rearrange this equation to solve for y :


y 2 = x 2 − 16
y = ± x 2 − 16 ​
Similar to equation 2, for a single value of x , there are two possible values of y (a positive and a negative square root), so this is not a function of x .

Conclusion Therefore, the equation that represents y as a function of x is x 2 = y .

Examples
In physics, the equation y = x 2 can describe the potential energy y of a spring compressed by a distance x . For each compression distance x , there is a unique potential energy y stored in the spring. This relationship is fundamental in understanding energy storage and release in mechanical systems.

Answered by GinnyAnswer | 2025-07-03

The only equation that represents y as a function of x is x 2 = y , since it yields a unique y value for each x value. Other equations provide multiple y values for given x , disqualifying them from being functions. Therefore, the correct answer is x 2 = y .
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Answered by Anonymous | 2025-07-04