Determine whether the following equation defines [tex]$y$[/tex] as a function of [tex]$x$[/tex].
[tex]$x-5=y^2$[/tex]
Does the equation [tex]$x-5=y^2$[/tex] define [tex]$y$[/tex] as a function of [tex]$x$[/tex]?
Answer (2)
The equation x − 5 = y 2 does not define y as a function of x because for each value of x greater than or equal to 5, there are two possible values for y . Thus, a single x can correspond to multiple y values. The answer is No.
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Solve the equation for y in terms of x : y = ± x − 5 .
Observe that for 5"> x > 5 , there are two values of y (positive and negative square roots).
Conclude that the equation does not define y as a function of x because a single x value can correspond to two y values.
The final answer is No, the equation does not define y as a function of x : N o .
Explanation
Understanding the Problem We are given the equation x − 5 = y 2 and asked to determine if it defines y as a function of x . In simpler terms, we need to check if for every value of x , there is only one corresponding value of y . If there's even one x that gives us multiple y values, then y is not a function of x .
Solving for y Let's solve the equation for y to see how y depends on x . Starting with x − 5 = y 2 , we can rewrite it as y 2 = x − 5 . To find y , we take the square root of both sides: y = ± x − 5
Analyzing the Solution Notice the ± sign in front of the square root. This means that for a single value of x , we can have two possible values for y . For example, if x = 6 , then y = ± 6 − 5 = ± 1 = ± 1 So, y can be either 1 or − 1 when x = 6 . This immediately tells us that y is not a function of x , because a function must have only one y value for each x value.
Conclusion Therefore, the equation x − 5 = y 2 does not define y as a function of x .
Examples
Consider a scenario where you're tracking the position of an object moving along a curve. If the equation relating the x and y coordinates of the object's position is such that for one x-coordinate, there are two possible y-coordinates, then the y-coordinate is not a function of the x-coordinate. This means you can't uniquely determine the object's vertical position based solely on its horizontal position. This concept is crucial in physics, engineering, and computer graphics, where understanding functional relationships is essential for modeling and predicting system behavior.
