Question

For the given functions f and g, find fog and state the domain of fog. Please show all of your work.
(see attached equation) For the given functions f and g, find fog and state the domain of fog. Please show all of your work.
(see attached equation) @Mathematics

Answer 1

fog means f(g(x)), so we solve the embedded function as such

\[f(x)=\sqrt{2x-5}\]\[g(x)=\left| x-1 \right|\]\[f(g(x))=\sqrt{2\left|x-1\right|-5}\]

in order to find domain you look at when that can be non-existent. since we have a square root, the arguement of the square root must be non-negative for the domain. so our domain is whenever the arguement is non-negative.
\[2\left|x-1\right|-5\ge0\]\[2\left|x-1\right|\ge5\]\[\left|x-1\right|\ge2.5\]\[-2.5\ge x-1\ge 2.5\]\[-1.5\ge x\ge 3.5\]or basically, whever x is NOT between -1.5 and 3.5

Answer 2

ok so the domain is \[1.5 \ge x \ge 3.5\] ?

Answer 3

-1.5

Answer 4

kinda, but you might want to write it as the following:
\[-1.5\ge x\] or \[x\ge 3.5\]
because x cannot satisfy both conditions at once

Answer 5

ok. thank you. :)

Answer 6

Can you help me with another one? I just posted it.