Determine whether the given equation represents y as a function of x and explain how you know.
a) 16x^2-y^2=0
b) 16-lxl=4y
thank you!
a) 16x^2-y^2=0
y^2 = 16 x^2
y = + 4 x or y = -4x
well y is not a unique function of x. There are two possible values of y for any old x
b) 16-lxl=4y
y = 4 - |x| /4
for every value of x there is one and only one value of y
(a) 16x^2 = y^2
not a function, since there are two values of y for almost all of the values of x.
Fails the vertical-line test.
(b) yes. each x has a unique |x|.
a) To determine whether the given equation represents y as a function of x, we need to check if for every value of x, there is a unique value of y.
In the equation 16x^2 - y^2 = 0, we can rearrange it to y^2 = 16x^2, and then take the square root of both sides to solve for y. We get y = ±4x.
Since for every value of x, there are two possible values for y (positive and negative), this equation does not represent y as a function of x. In a function, every input should have a unique output, but in this case, each input (x) has two possible outputs (y).
b) The equation 16 - |x| = 4y can be rearranged to isolate y:
| x | = 16 - 4y
|x| = 4(4 - y)
Now, we consider two cases:
Case 1: When 4 - y ≥ 0
In this case, the equation |x| = 4(4 - y) simplifies to x = 4(4 - y).
We can further rearrange it to y = 4 - (x/4).
Case 2: When 4 - y < 0
In this case, the equation |x| = 4(4 - y) simplifies to x = -4(4 - y).
We can rearrange it to y = -4 + (x/4).
Both cases provide a unique expression for y in terms of x. Hence, the equation represents y as a function of x.
To summarize, equation (a) does not represent y as a function of x since each x-value has two possible y-values. Equation (b), however, does represent y as a function of x since each x-value has a unique y-value.