Answer to Question #135576 in Algebra for Omar

Question

What can be said about the domain of the function:
f \circ g
where
f(y)= \frac{4}{y-2}
and
g(x)= \frac{5}{3x-1}
Express it in terms of a union of intervals of real numbers?
obtain the graph of f , g , and f \circ g .

Answer 1

Here, $f(y)=\dfrac{4}{y-2}\;and\;g(x)=\dfrac{5}{3x-1}.\\ \\$

$\mathbf{To\;find\;: Domain \;of\;f \circ g.}$

$\mathbf{Now,\;(f\circ g)(z)=f(g(z))=f\bigg(\dfrac{5}{3z-1}\bigg)=\dfrac{4}{\dfrac{5}{3z-1}-2} }$

$\mathbf{=\dfrac{4}{\dfrac{5-2(3z-1)}{3z-1}}=\dfrac{4(3z-1)}{5-6z+2}=\dfrac{12z-4}{7-6z}}$

$\mathbf{Clearly,\; f\circ g \; is\; defined\;on\;\R\; except\;when\; the\; denominator\;is\;zero,i.e.,\; when}$

$7-6z=0\;\implies z=\dfrac{7}{6}$

$\mathbf{So,\;domain\;of\;f\circ g=\R-\{{\dfrac{7}{6}}\}=(-\infty,\dfrac{7}{6})\cup(\dfrac{7}{6},-\infty)}$

Graph of f :-

Graph of g :-

Graph of f o g :-