Video Transcript
Find the domain of the function 𝑓
of 𝑥 equals the cubed root of 64 plus the square root of five 𝑥 minus one.
We recall that the domain of any
function 𝑓 of 𝑥 is the set of all possible values of 𝑥 such that 𝑓 of 𝑥 is
defined. In particular, we are looking for
any restrictions imposed by part of the function. The function we have been given is
not a simple one; it is a composition of a linear function within a square root
function within a cubed root function. Let’s consider what we know about
the domain of each type of function.
First, we recall that the domain of
the cube root function is the set of all real numbers. In other words, the cube root
function does not impose any domain restriction. The domain of the square root
function is all nonnegative real numbers. This means all real numbers in the
left-closed, right-open interval from zero to ∞ or that 𝑥 is greater than or equal
to zero. The domain of a linear function is
the set of all real numbers and thus imposes no restriction on the domain. Therefore, our only concern is the
square root part of the 𝑓 of 𝑥 function.
For a square root containing a
function, we will need to use a more general definition for the domain of the square
root of 𝑔 of 𝑥. The domain of the composite
function square root of 𝑔 of 𝑥 can be identified by finding the values of 𝑥
satisfying 𝑔 of 𝑥 greater than or equal to zero. In this case, 𝑔 of 𝑥 is five 𝑥
minus one. Therefore, we will find the domain
of 𝑓 of 𝑥 by letting five 𝑥 minus one be greater than or equal to zero.
To solve for 𝑥, we will first add
one to each side of the inequality. Then, we divide each side by five,
and we find that 𝑥 is greater than or equal to one-fifth. This means that 𝑓 of 𝑥 is defined
as long as all values of 𝑥 lie in the left-closed, right-open interval from
one-fifth to ∞.
