To determine which of the given expressions does not represent y as a function of x, let's analyze each one in order.
1. Equation 1: (x-3)^(2)-(y+4)^(2)=4
2. Equation 2: x^(2)-y=6x-5
3. Equation 3: 2x-3y=6
4. Equation 4: 4x^(7)=y-10
A function is defined as a relation where each input (or 'x' value) has exactly one output (or 'y' value). So we need to check each equation to see if there is any scenario where one x value can lead to two different y values.
For equation 1, we can rearrange to get:
(y+4)² = (x-3)² - 4
This can be simplified to two different results for y:
y = sqrt((x-3)² - 4) - 4
and
y = - sqrt((x-3)² - 4) - 4
Both equations provide a different value for y based on a single x input, indicating that equation 1 doesn't represent y as a function of x.
For equation 2, we can rearrange to get:
y = x² - 6x +5
Here, for every x input, we get exactly one y output. Therefore, it represents y as a function of x.
Similarly, for equation 3, after rearranging, we get:
y = (2/3)x - 2
Once again, each x value corresponds to one y value, so this equation represents y as a function of x.
Finally, for equation 4, after rearranging, we get:
y = 4x^7 + 10
This also provides one y value for each x input, implying that it also represents y as a function of x.
In conclusion, equation 1 is the ONLY equation which does not represent y as a function of x. Hence, the answer is 1.