What is the domain of the composite function #(g@f)(x)#?

Question

Sadie Alexander · Accepted Answer

Since the notation for the composition #(g\circ f)(x)=g(f(x))# means to first apply #f# to #x# to get #f(x)# and then to apply #g# to #f(x)# to get #g(f(x))#, a point #x# will be in the domain of #g\circ f# if and only if #x# is in the domain of #f# and #f(x)# is in the domain of #g#. For example, let #f(x)=\frac{x+3}{x-4}# and #g(x)=\frac{x-2}{x+5}#.  The number #x=4# is not in the domain of #f# and is therefore not in the domain of #g\circ f#.    Is there another number that's not in the domain of #g\circ f#?  Yes, any value(s) of #x# such that #f(x)=-5# is/are not in the domain of #g\circ f# since #-5# is not in the domain of #g#.  In order for #f(x)=-5#, we must have #x+3=-5(x-4)=-5x+20# or #6x=17# or #x=17/6#.  The domain of #g\circ f# is therefore #\{x\in R: x!= 4 \mbox{ and } x!=17/6\}#. If you find a simplified formula for #g\circ f# in such an example, you can be misled in what the correct answer is.  In the example above, we can write #(g\circ f)(x)=\frac{f(x)-2}{f(x)+5}=\frac{\frac{x+3}{x-4}-2}{\frac{x+3}{x-4}+5}=\frac{x+3-2(x-4)}{x+3+5(x-4)}=\frac{-x+11}{6x-17}#.   This simplified formula can mislead you into thinking that #17/6# is the only number not in the domain of #g\circ f#, but it's not.  The number #4# is also not in the domain as we saw above. Why does this happen?  The reason is that the equality #\frac{\frac{x+3}{x-4}-2}{\frac{x+3}{x-4}+5}=\frac{x+3-2(x-4)}{x+3+5(x-4)}# is not true if #x=4#, because the expression on the left-hand side is undefined there.   So, in doing the simplification above, we were implicitly assuming that #x!=4#.

Lily Anderson · Answer

undefined To determine the domain of the composite function $(g \circ f)(x)$, we need to consider the domains of the individual functions involved and any restrictions they impose on the input values.

Let's denote $f(x)$ as the inner function and $g(x)$ as the outer function.

First, we find the domain of $f(x)$, considering any restrictions on the input values imposed by this function.

Next, we need to ensure that the output of $f(x)$ falls within the domain of $g(x)$ so that we can evaluate $(g \circ f)(x)$. This means we need to check if the output of $f(x)$ is within the domain of $g(x)$.

The domain of $(g \circ f)(x)$ will be the set of all $x$ values for which both $f(x)$ and $g(x)$ are defined and where the output of $f(x)$ lies within the domain of $g(x)$.

In summary, to find the domain of $(g \circ f)(x)$, we need to:

1. Determine the domain of $f(x)$.
2. Determine the range of $f(x)$ within the domain of $g(x)$.
3. Ensure that the range of $f(x)$ within the domain of $g(x)$ is also within the domain of $g(x)$.
4. The resulting set of values will be the domain of the composite function $(g \circ f)(x)$.