i) xy=100
ii) 9x^2+y^2=16
Are these both function or nor a function? if so then how?
i) is a function, because each distinct x corresponds to a distinct y.
ii) is not, because there is a + and a - value for y for each value of x.
To determine if these equations represent functions or not, we need to examine the relationship between the variables x and y in each equation.
i) xy = 100
In this equation, we have a product of two variables, x and y, equal to a constant, 100. This equation does not provide a direct relationship between x and y, but rather represents all possible pairs of x and y that satisfy the equation.
To determine if this equation represents a function, we can check if each value of x corresponds to a unique value of y. We can do this by examining the x and y values that satisfy the equation.
If we consider x = 4, we can solve the equation for y by substituting the value of x:
4y = 100
Dividing both sides by 4, we get:
y = 25
Thus, for x = 4, y = 25.
Similarly, if we consider x = 5, we have:
5y = 100
y = 20
So for x = 5, y = 20.
By finding these correspondences for different values of x, we can see that each value of x does indeed correspond to a unique value of y. Therefore, the equation xy = 100 is a function.
ii) 9x^2 + y^2 = 16
In this equation, we have a sum of two variables, 9x^2 and y^2, equal to a constant, 16. Similar to the first equation, this equation does not provide a direct relationship between x and y.
To check if it represents a function, we can follow the same approach as before. By solving for y, we can see if each value of x corresponds to a unique value of y. However, when we solve for y in terms of x, we encounter a problem. Taking the square root on both sides of the equation introduces a ± sign, indicating two possible values of y for each x.
For example, if we choose x = 1, we have:
9(1)^2 + y^2 = 16
9 + y^2 = 16
y^2 = 7
y = ±√7
Since for x = 1, we have two potential values of y, ±√7, the equation 9x^2 + y^2 = 16 does not represent a function.
In summary, the first equation, xy = 100, represents a function, whereas the second equation, 9x^2 + y^2 = 16, does not represent a function.