Which equation is a function of $x$?
$x=5$
$x-y^2+9$
$x^2=y$
$x^2-y^2+16$
Answer (1)
Analyze each equation to determine if it defines y as a function of x .
x = 5 is not a function of x because x is constant.
x − y 2 + 9 = 0 is not a function of x because for each x there are two y values.
x 2 − y 2 + 16 = 0 is not a function of x because for each x there are two y values.
x 2 = y is a function of x because for each x there is only one y value. Therefore, the answer is x 2 = y .
Explanation
Understanding the Problem We are given four equations and asked to identify which one represents y as a function of x . An equation represents y as a function of x if for every value of x in its domain, there exists exactly one corresponding value of y . We will analyze each equation to determine if it meets this criterion.
Analyzing the first equation
x = 5 : This equation represents a vertical line where x is always 5, regardless of the value of y . Therefore, for x = 5 , y can be any real number. This does not define y as a function of x .
Analyzing the second equation
x − y 2 + 9 = 0 : We can rewrite this equation as y 2 = x + 9 . Taking the square root of both sides, we get y = ± x + 9 . For -9"> x > − 9 , there are two possible values of y for each value of x . For example, if x = 0 , then y = ± 9 = ± 3 . Since there are two y values for a single x value, this is not a function of x .
Analyzing the third equation
x 2 = y : This equation can be rewritten as y = x 2 . For every value of x , there is only one value of y (the square of x ). Therefore, this equation defines y as a function of x .
Analyzing the fourth equation
x 2 − y 2 + 16 = 0 : We can rewrite this equation as y 2 = x 2 + 16 . Taking the square root of both sides, we get y = ± x 2 + 16 . For every value of x , there are two possible values of y . For example, if x = 0 , then y = ± 16 = ± 4 . Since there are two y values for a single x value, this is not a function of x .
Conclusion Based on our analysis, the only equation that represents y as a function of x is x 2 = y .
Examples
In physics, the relationship between the distance an object falls ( y ) and the time it has been falling ( x ) can be modeled by the equation y = k x 2 , where k is a constant. This is a function because for each moment in time, the object has fallen a specific distance. Understanding functional relationships like this is crucial for predicting the behavior of objects in motion.
