Question

For the given functions f, g and h, find f○g○h and state the exact domain of f○g○h. Please show all of your work.
f(x) = 1/x
g(x) - lnx
h(x) = 2x+15

Answer 1

h(x) = 2x + 15
g(h(x)) = - ln (2x+15)
So what's f(g(h(x)))?

Answer 2

\[h(x)=2x+15 \text{ domain of } h: \mathbb{R}\]
\[g(x)=-\ln(x) \text{ domain of }g =\{x \in \mathbb{R}: x>0\}\]
\[g\circ h(x)=-\ln(2x+15) \text{ domain of }g \circ h =\{x \in \mathbb{R}: 2x+15>0\}=\{x \in \mathbb{R}: x>-\frac{15}{2}\}\]
\[f(x)=\frac{1}{x} \text{ domain of } f=\{x \in \mathbb{R}: x \neq 0\}\]
\[f \circ g \circ h(x)=-\frac{1}{\ln(2x+15)} \text{ domain of } f \circ g \circ h =\{x \in \mathbb{R}: 2x+15>0 \text{ and } \ln(2x+15) \neq 0\}\]
\[\ln(2x+15)\neq0 \Rightarrow 2x+15 \neq 1 \Rightarrow x \neq -7\]
\[\text{ domain of } f \circ g \circ h =\{x \in \mathbb{R}: x \in ]-\frac{15}{2},-7[ \cup]-7,+\infty[ \}\]

Answer 3

vood is this the answer?