Given the function below,
[tex]f(x)=\frac{\sqrt[]{9-7x}}{6x^2+13x-15}[/tex]To find the domain of the function above, we factorise the denominators to find x factors,
[tex]\begin{gathered} \text{Given the denominator,} \\ 6x^2+13x-15 \end{gathered}[/tex]Factorising the denominator,
[tex]\begin{gathered} 6x^2+13x-15=0 \\ 6x^2+18x-5x-15=0 \\ 6x(x+3)-5(x+3)=0 \\ (6x-5)(x+3)=0 \\ x+3=0 \\ x=-3 \\ 6x-5=0 \\ 6x=5 \\ x=\frac{5}{6} \\ x=-3\text{ or }\frac{5}{6} \end{gathered}[/tex]The domain of a function is the set of all possible values of x, except for x=0.
Hence, the domain of the function is the set of all possible values of x except -3 and 5/6.
