Evaluate the piecewise function at the given value of the independent variable. \[ g(x)=\left\{\begin{array}{ll} \frac{x^{2}+6}{x-4} & \text { if } x \neq 4 \\ x+2 & \text { if } x=4 \end{array} ; g(7)\right. \] (A) 9 (B) 7 (C) $\frac{13}{3}$ (D) $\frac{55}{3}$

Step 1 ：The given function is a piecewise function, which means it is defined by different expressions for different ranges of the independent variable, x.

Step 2 ：We are asked to evaluate the function g(x) at x = 7.

Step 3 ：Looking at the definition of the function, we see that when x ≠ 4, g(x) = \(\frac{x^2 + 6}{x - 4}\). Since 7 ≠ 4, we use this expression to evaluate g(7).

Step 4 ：Substitute x = 7 into the expression: g(7) = \(\frac{7^2 + 6}{7 - 4} = \frac{49 + 6}{3} = \frac{55}{3}\)

Step 5 ：So, the value of g(7) is \(\frac{55}{3}\).

Step 6 ：Therefore, the correct answer is \(\boxed{\frac{55}{3}}\).