Composite Functions

Question

$f(x)= \dfrac{1}{10x+17}+13$

$g(x)= \dfrac{1}{9x-6}$

I need to find $f(g(x)).$

How do I do this? I keep on getting it wrong. The correct answer is $\dfrac{1998x-1202}{153x-92}$. But I am unsure how to get to this. Thankyou.

Answer 1

Wherever $x$ appears in $f(x)$, plug in the entire function $g(x)$:

$$f(g(x)) = \frac{1}{10(g(x)) + 17} + 13 = \frac{1}{10\left(\dfrac{1}{9x-6}\right) + 17} + 13 $$

Multiply numerator and denominator of the main fraction by $9x-6$:

$$f(g(x))= \frac{9x-6}{(9x-6)\cdot 10\left(\dfrac{1}{9x-6}\right) + 17} + 13 $$ $$=\frac{9x-6}{10\left(1\right) + 17(9x-6)} + 13 $$

Now, simplify, to get $$f(g(x))= \frac{9x - 6}{153x-92} + \frac{13(153x - 92)}{153-92}$$

Can you take it from here to complete the final addition, and simplify the numerator?