Answer
Domain: $\{x|x\ne5\}$
Work Step by Step
Since functions are defined only over the set of real numbers, then the denominator of a rational function must not be zero. Thus, the denominator of the given function, $ g(x)=\dfrac{x-4}{x^2-10x+25} ,$ should be nonzero. Solving for the value/s of the variable that will make the denominator equal to zero results to, \begin{align*} x^2-10x+25&=0 \\ (x-5)^2&=0 &(\text{use factoring of perfect square trinomials}) \\ x-5&=\pm0 &(\text{take square root of both sides}) \\ x-5&=0 \\ x-5+5&=0+5 \\ x&=5 .\end{align*} The solution above shows that if $x=5,$ the denominator of the given function becomes zero (which is not allowed). Hence, the domain of $g(x)=\dfrac{x-4}{x^2-10x+25}$ is $\{x|x\ne5\}$.
