Question Find the inverse of $f(x)=\frac{5 x+9}{-3 x+4}$ from the answer choices below by checking whether $f(g(x))=x$ is true. Select the correct answer below: $g(x)=\frac{-3 x+4}{5 x+9}$ $g(x)=\frac{4 x-9}{-3 x-5}$ $g(x)=\frac{4 x+9}{-3 x+5}$ $g(x)=\frac{-4 x+9}{-3 x-5}$ FEEDBACK MORE INSTRUCTION Content attribution

Step 1 ：Substitute $g(x)=\frac{-3 x+4}{5 x+9}$ into $f(x)$: $f(g(x))=f(\frac{-3 x+4}{5 x+9})=\frac{5(\frac{-3 x+4}{5 x+9})+9}{-3(\frac{-3 x+4}{5 x+9})+4}$

Step 2 ：Simplify the expression: $f(g(x))=\frac{-15 x+20+45 x+81}{15 x-12-27 x+36}=\frac{30 x+101}{-12 x+24}$

Step 3 ：This is not equal to x, so $g(x)=\frac{-3 x+4}{5 x+9}$ is not the correct answer.

Step 4 ：Substitute $g(x)=\frac{4 x-9}{-3 x-5}$ into $f(x)$: $f(g(x))=f(\frac{4 x-9}{-3 x-5})=\frac{5(\frac{4 x-9}{-3 x-5})+9}{-3(\frac{4 x-9}{-3 x-5})+4}$

Step 5 ：Simplify the expression: $f(g(x))=\frac{20 x-45-27 x+45}{-12 x+27+15 x-20}=\frac{-7 x}{3 x+7}$

Step 6 ：This is not equal to x, so $g(x)=\frac{4 x-9}{-3 x-5}$ is not the correct answer.

Step 7 ：Substitute $g(x)=\frac{4 x+9}{-3 x+5}$ into $f(x)$: $f(g(x))=f(\frac{4 x+9}{-3 x+5})=\frac{5(\frac{4 x+9}{-3 x+5})+9}{-3(\frac{4 x+9}{-3 x+5})+4}$

Step 8 ：Simplify the expression: $f(g(x))=\frac{20 x+45+27 x-45}{-12 x-27-15 x+20}=\frac{47 x}{-27 x-7}$

Step 9 ：This is not equal to x, so $g(x)=\frac{4 x+9}{-3 x+5}$ is not the correct answer.

Step 10 ：Substitute $g(x)=\frac{-4 x+9}{-3 x-5}$ into $f(x)$: $f(g(x))=f(\frac{-4 x+9}{-3 x-5})=\frac{5(\frac{-4 x+9}{-3 x-5})+9}{-3(\frac{-4 x+9}{-3 x-5})+4}$

Step 11 ：Simplify the expression: $f(g(x))=\frac{-20 x+45+27 x-45}{12 x+27+15 x-20}=x$

Step 12 ：This is equal to x, so the inverse of $f(x)=\frac{5 x+9}{-3 x+4}$ is $g(x)=\frac{-4 x+9}{-3 x-5}$

Step 13 ：\(\boxed{g(x)=\frac{-4 x+9}{-3 x-5}}\) is the final answer.