Determine whether the following equation defines y as a function of x.
[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]?
Yes
No
Answer (2)
The equation x − 5 = y 2 does not define y as a function of x because for certain values of x greater than 5, there are two possible values of y (one positive and one negative). Therefore, the chosen option is N o .
;
Solve the equation for y in terms of x : y = ± x − 5 .
Observe that for a single value of x , there are two possible values of y .
Conclude that the equation does not define y as a function of x because a function must have a unique y value for each x value.
The final answer is N o .
Explanation
Analyze the problem We are given the equation x − 5 = y 2 and asked to determine if it defines y as a function of x . A function must have a unique y value for each x value. To determine if this is the case, we can solve the equation for y in terms of x .
Solve for y Solving for y , we get:
y 2 = x − 5
y = p m s q r t x − 5
Find y values for a given x This means that for a single value of x , there can be two possible values of y , one positive and one negative. For example, if x = 9 , then
y = p m s q r t 9 − 5 = p m s q r t 4 = p m 2
So, when x = 9 , y can be 2 or − 2 .
Conclusion Since there are two possible y values for a single x value (when 5"> x > 5 ), the equation does not define y as a function of x .
Examples
In physics, understanding functional relationships is crucial. For example, the equation x − 5 = y 2 could represent a simplified relationship between the position ( x ) of an object and its velocity ( y ). However, because y is not a function of x here, it implies that knowing the position alone isn't enough to uniquely determine the velocity; the object could be moving in either direction at that position. This highlights the importance of understanding whether a relationship is truly functional in making accurate predictions.
