Question

For the given functions f, g, and h, find f o g o h and state the exact domain of .
f(x)=1/x, g(x)=lnx, h(x)=2x+15

Answer 1

\[h(x)=2x+5\], domain is all real numbers but
\[g(x)=\ln(x)\] so domain is
\[x>0\] now
\[g\circ h(x)=g(h(x))=g(2x+5)=\ln(2x+5)\] so now for domain we have
\[2x+5>0\] solving gives
\[2x>-5,x>-\frac{5}{2}\] for our new domain

Answer 2

finally
\[f(x)=\frac{1}{x}\] and domain is all numbers except 0, so
\[f\circ g\circ h(x)=f\circ \ln(2x+5)=\frac{1}{\ln(2x+5)}\] and now
\[\ln(2x+5)\] cannot be zero. that means
\[2x+5\neq 1\] because
\[\ln(1)=0\] and so we have to solve
\[2x+5=1\]
\[x=-2\] so domain is
\[-\frac{5}{2}<x, x\neq -2\]