select the correct answer. rational functions v and w both have a point of discontinuity at x = 7. which…

Explanation:

Step1: Recall function - transformation rules

For a function (y = f(x)), a horizontal shift of (h) units to the right is given by (y=f(x - h)), and a horizontal shift of (h) units to the left is given by (y = f(x+h)), and a vertical shift of (k) units up is given by (y=f(x)+k). The point of discontinuity of a rational - function is related to the values that make the denominator zero.

Step2: Analyze the effect of each transformation on the point of discontinuity

If the function (v(x)) has a point of discontinuity at (x = 7), we want to find a transformation of (v(x)) such that the new function (w(x)) also has a point of discontinuity at (x = 7). For option A: If (w(x)=v(x - 7)), the point of discontinuity of (v(x)) which is at (x = 7) for (v(x)) will be shifted 7 units to the right. For (v(x)), when (x = 7) is the point of discontinuity. For (w(x)=v(x - 7)), we set (x-7 = 7), then (x=14) is the point of discontinuity. For option B: If (w(x)=v(x + 7)), the point of discontinuity of (v(x)) which is at (x = 7) for (v(x)) will be shifted 7 units to the left. We set (x + 7=7), then (x = 0) is the point of discontinuity. For option C: If (w(x)=v(x - 7)+7), first, the function (v(x)) is shifted 7 units to the right (so the point of discontinuity of (v(x)) at (x = 7) moves to (x = 14)) and then shifted 7 units up. The vertical shift does not affect the (x) - value of the point of discontinuity, but the horizontal shift moves it to (x = 14). For option D: If (w(x)=v(x)+7), this is a vertical shift of the function (v(x)) by 7 units up. A vertical shift does not change the (x) - value of the point of discontinuity. Since (v(x)) has a point of discontinuity at (x = 7), (w(x)=v(x)+7) will also have a point of discontinuity at (x = 7).

Answer:

D. (w(x)=v(x)+7)