Final answer:
Option A is the correct answer because by subtracting 7 from the x-value inside the function w(x), it shifts the graph of v(x) to the right by 7 units, preserving the point of discontinuity at x = 7.
Step-by-step explanation:
A rational function is a function represented by the ratio of two polynomials. The point of discontinuity at x = 7 means that the function is not defined at that specific value of x. In order to have the same point of discontinuity for function w as the given function v, the transformation applied to v must have the effect of shifting the discontinuity along the x-axis, without altering its position.
With transformations, if you add or subtract a number outside the function, it translates the graph up or down, which wouldn’t affect the x-coordinate of the discontinuity. Therefore, option D would simply shift the graph of v(x) vertically and wouldn't affect the x-coordinate of the point of discontinuity. Option C shifts v(x) vertically as well and therefore also won't affect the point of discontinuity at x = 7.
In contrast, adding or subtracting inside the function's argument moves the graph left or right along the x-axis, thereby affecting the x-coordinate of the point of discontinuity. Option B, w(x) = v(x + 7), shifts the function v(x) to the left by 7 units, which moves the discontinuity from x = 7 to x = 0. Therefore, option B would not result in the same discontinuity.
Choosing option A, w(x) = v(x - 7), also affects the horizontal positioning of discontinuities. This transformation shifts the graph of v(x) to the right by 7 units, which keeps the discontinuity of function w at x = 7. Hence, option A is correct.