Determine whether the following equation defines y as a function of x. [tex]x^2+y^2=121[/tex]
Does the equation [tex]x^2+y^2=121[/tex] define [tex]y[/tex] as a function of [tex]x[/tex]?
Yes
No
Answer (2)
The equation x 2 + y 2 = 121 does not define y as a function of x because for one value of x , there are two possible values of y : one positive and one negative. Therefore, the answer is No: oxed{No} .
;
Solve the equation for y in terms of x : y = ± 121 − x 2 .
Observe that for a single value of x , there are two possible values of y (positive and negative).
Conclude that the equation does not define y as a function of x .
The final answer is No: N o .
Explanation
Understanding the Problem We are given the equation x 2 + y 2 = 121 and asked to determine if it defines y as a function of x . A function requires that for each value of x , there is only one corresponding value of y . We need to solve the equation for y and see if there are multiple possible values of y for a given x .
Solving for y To determine if the equation defines y as a function of x , we solve for y in terms of x :
x 2 + y 2 = 121
Subtract x 2 from both sides:
y 2 = 121 − x 2
Take the square root of both sides:
y = { p m } 121 − x 2
Checking for Multiple y Values The equation y = ± 121 − x 2 shows that for a single value of x , there are two possible values of y , one positive and one negative (unless y = 0 ). For example, if x = 0 , then y = ± 121 − 0 2 = ± 121 = ± 11 . So, when x = 0 , y can be either 11 or − 11 . Since there are two possible y values for a single x value, the equation does not define y as a function of x .
Conclusion Since there are two possible values of y for a single value of x , the equation x 2 + y 2 = 121 does not define y as a function of x .
Examples
Consider a circular garden bed with a radius of 11 feet. The equation x 2 + y 2 = 121 describes this circle. If we want to find the height ( y ) of the garden bed at a certain horizontal distance ( x ) from the center, we realize there are two possible heights (positive and negative) for most x values. This means knowing the horizontal distance alone isn't enough to uniquely determine the height, illustrating why the equation doesn't represent y as a function of x . This concept applies in various scenarios, such as designing circular structures or analyzing data points on a circular path.
