Determine whether the following equation defines y as a function of x.
[tex]x^2+y=16[/tex]
Does the equation [tex]x^2+y=16[/tex] define [tex]y[/tex] as a function of [tex]x[/tex]?
Yes
No
Answer (2)
The equation x 2 + y = 16 defines y as a function of x because solving for y gives y = 16 − x 2 , which produces a unique value of y for every x . Thus, the answer is Y es .
;
Solve the equation for y : y = 16 − x 2 .
Verify that for each x , there is only one y value.
Conclude that the equation defines y as a function of x .
The equation x 2 + y = 16 defines y as a function of x : $\boxed{Yes}.
Explanation
Understanding the Problem We are given the equation x 2 + y = 16 and we want to determine if this equation defines y as a function of x . In other words, we want to see if for every value of x , there is only one corresponding value of y .
Solving for y To determine if y is a function of x , we need to solve the equation for y in terms of x . We can do this by subtracting x 2 from both sides of the equation: y = 16 − x 2 Now we have y expressed as a function of x .
Checking for Uniqueness For any value of x , the expression 16 − x 2 will result in a single, unique value for y . This is because squaring a number gives a unique result, and subtracting that result from 16 also gives a unique result. Therefore, for each value of x , there is only one corresponding value of y .
Conclusion Since each value of x corresponds to only one value of y , the equation x 2 + y = 16 defines y as a function of x .
Examples
In physics, the equation x 2 + y = 16 can represent a constraint on the possible values of two related physical quantities, x and y . For example, x could be the position of an object along one axis, and y could be related to its potential energy. Knowing that y is a function of x allows us to predict the potential energy of the object based on its position. This concept is crucial in understanding how systems evolve and conserve energy.
