Answer:
C. The [tex]x[/tex]-value of 4 produces the same [tex]y[/tex]-value in both [tex]f(x) = (x-2)^{3}-6[/tex] and [tex]g(x) = \sqrt[3]{x+4}[/tex].
Step-by-step explanation:
Let [tex]f(x) = (x-2)^{3}-6[/tex] and [tex]g(x) = \sqrt[3]{x+4}[/tex]. A value of [tex]x[/tex] is a solution of the formula if and only if [tex]f(x) = g(x)[/tex]. If we know that [tex]x = 4[/tex], then values of each function are, respectively:
[tex]f(x) = (x-2)^{3}-6[/tex]
[tex]f(4) = 2^{3}-6[/tex]
[tex]f(4) = 2[/tex]
[tex]g(x) = \sqrt[3]{x+4}[/tex]
[tex]g(4) = \sqrt[3]{8}[/tex]
[tex]g(4) = 2[/tex]
Since [tex]f(4) = g(4)[/tex], then [tex]x = 4[/tex] is a solution of [tex](x-2)^{3}-6 = \sqrt[3]{x+4}[/tex]. Hence, correct answer is C.
