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We have to try the answer choices in this case to see which of them fits the bill.
A.
g(x) = x^2. (We then apply this to g(c), to g(d) and to g(c-d) by simply putting c, d, or c-d in place of the x in the original function definition.
g(c) = c^2
g(d) = d^2
g(c) - g(d) = c^2 - d^2
g(c-d) = (c-d)^2 = c^2 - 2cd + d^2 This is not equal to the line above, so this isn't our answer .
B.
g(x) = x+5
g(c) = c+5
g(d) = d+5
g(c) - g(d) = c+5 - (d+5) = c + 5 - d - 5 = c-d
g(c-d) = c-d+5 (Again, this isn't equal to the line above, so not our answer.)
C. g(x) = sqrt3 * x
g(c) = sqrt3c
g(d) = sqrt3d
g(c) - g(d) = sqrt3c - sqrt3d
g(c-d) = sqrt3(c-d) This one isn't equal to the line above, so isn't our answer.
D. g(x) = 5x
g(c) = 5c
g(d) = 5d
g(c) - g(d) = 5c - 5d = 5*(c-d)
g(c-d) = 5*(c-d) This one IS equal to the line above, so is our answer.
E. g(x) = 15/x
g(c) = 15/c
g(d) = 15/d
g(c) - g(d) = 15/c - 15/d
g(c-d) = 15/(c-d) Again, not equal to the line above, so not our answer.
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