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) $\frac{55}{3}$
Answer
Therefore, the correct answer is \(\boxed{\frac{55}{3}}\).
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 \(\boxed{\frac{55}{3}}\).
