Let's analyze the given options to determine which equation for [tex]\( w(x) \)[/tex] could have a point of discontinuity at [tex]\( x = 7 \)[/tex], matching the behavior of [tex]\( v(x) \)[/tex].
### Option A: [tex]\( w(x) = v(x - 7) \)[/tex]
For [tex]\( w(x) \)[/tex] to have a discontinuity at [tex]\( x = 7 \)[/tex], we set [tex]\( x = 7 \)[/tex] in the equation:
[tex]\[ w(7) = v(7 - 7) = v(0) \][/tex]
This means that [tex]\( w \)[/tex] will be discontinuous at [tex]\( x = 7 \)[/tex] if [tex]\( v \)[/tex] is discontinuous at [tex]\( x = 0 \)[/tex]. Since [tex]\( v \)[/tex] has a discontinuity at [tex]\( x = 7 \)[/tex], we shift the input by 7 to produce a discontinuity at [tex]\( 7 - 7 = 0 \)[/tex]. Thus, [tex]\( v(0) \)[/tex] should be undefined or discontinuous.
### Option B: [tex]\( w(x) = v(x + 7) \)[/tex]
For [tex]\( w(x) \)[/tex] to have a discontinuity at [tex]\( x = 7 \)[/tex], we set [tex]\( x = 7 \)[/tex] in the equation:
[tex]\[ w(7) = v(7 + 7) = v(14) \][/tex]
This means [tex]\( w \)[/tex] would be discontinuous at [tex]\( x = 7 \)[/tex] if [tex]\( v \)[/tex] is discontinuous at [tex]\( x = 14 \)[/tex]. However, since [tex]\( v \)[/tex] specifically has a discontinuity at [tex]\( x = 7 \)[/tex], this transformation doesn't introduce a discontinuity at [tex]\( x = 7 \)[/tex].
### Option C: [tex]\( w(x) = v(x - 7) + 7 \)[/tex]
For [tex]\( w(x) \)[/tex] to have a discontinuity at [tex]\( x = 7 \)[/tex], we set [tex]\( x = 7 \)[/tex] in the equation:
[tex]\[ w(7) = v(7 - 7) + 7 = v(0) + 7 \][/tex]
This means [tex]\( w \)[/tex] would be discontinuous at [tex]\( x = 7 \)[/tex] if [tex]\( v \)[/tex] is discontinuous at [tex]\( x = 0 \)[/tex], similar to Option A, but with an added constant 7. While [tex]\( v(0) \)[/tex] could be undefined, adding 7 does not affect the discontinuity of [tex]\( v \)[/tex] itself, which means the discontinuity at [tex]\( x = 7 \)[/tex] in [tex]\( w \)[/tex] is still effectively dependent on [tex]\( v(0) \)[/tex].
### Option D: [tex]\( w(x) = v(x) + 7 \)[/tex]
For [tex]\( w(x) \)[/tex] to have a discontinuity at [tex]\( x = 7 \)[/tex], we consider:
[tex]\[ w(7) = v(7) + 7 \][/tex]
In this case, the value of [tex]\( v(7) \)[/tex] directly affects the discontinuity. Since [tex]\( v(x) \)[/tex] is given to have a discontinuity at [tex]\( x = 7 \)[/tex], this equation would maintain the same discontinuity for [tex]\( w \)[/tex].
Given these analyses, we find that the proper transformation which shifts the input to produce a new discontinuity at [tex]\( x = 7 \)[/tex] mirroring the original [tex]\( x = 7 \)[/tex] in [tex]\( v \)[/tex] is:
Option A: [tex]\( w(x) = v(x - 7) \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{A}
\][/tex]
