Step 1 :Define the functions $f(x)=6x+5$ and $g(x)=4x-7$.
Step 2 :To find $(f+g)(x)$, add the functions $f(x)$ and $g(x)$ together to get $10x - 2$.
Step 3 :To find $(f-g)(x)$, subtract $g(x)$ from $f(x)$ to get $2x + 12$.
Step 4 :To find $(fg)(x)$, multiply $f(x)$ and $g(x)$ together to get $24x^2 - 2x - 35$.
Step 5 :To find $\left(\frac{f}{g}\right)(x)$, divide $f(x)$ by $g(x)$ to get $\frac{6x + 5}{4x - 7}$.
Step 6 :The domain of $\frac{f}{g}$ is all real numbers except for the values of $x$ that make $g(x) = 0$, because division by zero is undefined. Solve $4x - 7 = 0$ to find that $x \neq \frac{7}{4}$.
Step 7 :Final Answer: $(f+g)(x)=\boxed{10x - 2}$, $(f-g)(x)=\boxed{2x + 12}$, $(f g)(x)=\boxed{24x^2 - 2x - 35}$, $\left(\frac{f}{g}\right)(x)=\boxed{\frac{6x + 5}{4x - 7}}$, The domain of $\frac{f}{g}$ is $\boxed{x \neq \frac{7}{4}}$