Video Transcript
If the function π of π₯ equals two
over π₯, where π₯ is not equal to zero, and the function π of π₯ equals π₯ minus
41, determine the domain of π of π.
We recall first that this notation
of π and then a little circle and then π means π of π. Itβs the composite function π of
π of π₯. This is the composite function we
get when we take an input, apply the function π, and then apply the function π to
the result. Weβre given the definitions of the
two functions π and π in the question and asked to determine the domain of this
composite function.
Now we recall that the domain of a
function is the set of all input values to that function. Or we can think of it as the set of
all values on which the function acts. We should notice that the domain of
the function π of π₯ is actually given in the question; itβs π₯ not equal to
zero. And this is because in the function
π, we are dividing two by π₯. And because division by zero is
undefined, the function π canβt take the value zero as an input, but it can take
any other real value.
Remember though we want to find the
domain of the composite function π of π of π₯. So, letβs begin by finding an
algebraic expression for π of π of π₯. If we start with an input π₯ and
then apply the function π first of all, we have π of π₯ is equal to π₯ minus
41. We then need to apply the function
π, which means we take π of π₯ β thatβs π₯ minus 41 β as our input to the function
π. So, we have π of π₯ minus 41. π is the function that divides two
by its input value. So, π of π₯ minus 41 is two over
π₯ minus 41.
We have then an algebraic
expression for the composite function π of π of π₯, and we can now consider its
domain. This expression will be well
defined, provided the denominator is not equal to zero. In other words, it will be
undefined if the denominator is equal to zero, so if π₯ minus 41 is equal to zero,
which leads to π₯ equals 41. This means that the function canβt
act on the value 41 because it would lead to two divided by zero, which is
undefined. The function can act on all other
real values of π₯.
So, the domain of the composite
function π of π is the set of all real numbers minus the value 41.
