Video Transcript
Which of the following functions
are equal?
We have been given five pairs of
rational functions to consider. For each option, the 𝑛 sub one
function equals one over 𝑥. To begin, we will recall the
requirements for two rational functions to be considered equal.
The first requirement for 𝑛 sub
one and 𝑛 sub two to be equal is for their domains to be equal. In general, the domain of a
rational function is the set of all real numbers minus the set containing the zeros
of the denominator. That means for 𝑛 sub one and 𝑛
sub two to have equal domains, they must have the same zeros of the denominator. The second requirement is that 𝑛
sub one must be equivalent to 𝑛 sub two on their shared domain. Equivalent functions simplify to
give the same expression.
In summary, to show two functions
are equal, they must have the same domain and simplify to the same expression.
Let’s clear some space to work out
this problem. First, we will find the domain of
the 𝑛 sub one function. The domain of a rational function
is the set of all real numbers minus the set containing the zeros of the
denominator. We find the zeros or solutions of
the denominator by setting it equal to zero and solving for 𝑥. In this case, 𝑥 equals the real
number zero. Thus, the domain of 𝑛 sub one is
the set of all real numbers minus the set containing zero.
Now we will look at the first 𝑛
sub two function, which equals 𝑥 minus 53 over 𝑥 squared minus 53𝑥. We set the denominator equal to
zero to find the set of real numbers to exclude from the domain. To solve for 𝑥, we will factor the
quadratic expression 𝑥 squared minus 53𝑥. The factors are 𝑥 and 𝑥 minus
53. Then we set each factor equal to
zero to solve, which gives us two solutions: zero and 53. These are the values to exclude
from our domain. So the domain of the 𝑛 sub two
function given in option (A) is the set of all real numbers minus the set containing
zero and 53. The domain of 𝑛 sub one does not
exclude the number 53, so we eliminate option (A) because the domains are not
equal.
However, if we first simplified the
𝑛 sub two function, we might have expected it to equal 𝑛 sub one. The 𝑛 sub two function simplifies
to one over 𝑥, but this fact alone does not make it equal to the 𝑛 sub one
function because the domains are not the same.
Moving on to option (B), we again
set the denominator equal to zero. Then we factor the cubic expression
𝑥 cubed minus 53𝑥. We set both factors, the 𝑥 and the
𝑥 squared minus 53, equal to zero. This leads to three real number
solutions: zero, the square root of 53, and the negative square root of 53. These are the real numbers excluded
from the domain. This is not equal to the domain of
the 𝑛 sub one function, so we eliminate choice (B). We can show that the 𝑛 sub two
function from choice (B) equals one over 𝑥, but this does not change our
conclusion.
Next, we find the domain of the 𝑛
sub two function in option (C). We factor the denominator and set
each of those factors equal to zero. We have 𝑥 equals zero and 𝑥
squared plus 53 equals zero. When we solve the second equation,
we come up with the positive and negative square root of negative 53, which has no
real number values. Therefore, the only real number
value that is eliminated from our domain is zero. This means we have found an 𝑛 sub
two function with the same domain as our 𝑛 sub one function. But we still need to check the
second requirement for the equality of functions.
We rewrite the denominator of the
𝑛 sub two function in its factored form. We notice that the numerator and
the denominator share the factor 𝑥 squared plus 53, so we cancel. This leaves us with the simplified
expression one over 𝑥. We have just shown that these two
functions are equal because their domains are equal, and the 𝑛 sub two function
simplifies to an expression that is equal to the 𝑛 sub one function.
Let’s check the last two pairs of
functions.
For option (D), we set the
denominator of the 𝑛 sub two function equal to zero. The only 𝑥-value that satisfies
the equation 𝑥 squared equals zero is the real number zero. This means that we have found that
the domain of 𝑛 sub two is the same as the domain of 𝑛 sub one, that is, the set
of all real numbers minus the set containing zero.
The problem with option (D) occurs
when we go to simplify the 𝑛 sub two function. The factorization of the numerator
is 𝑥 times 𝑥 plus 53. And the factorization of the
denominator is 𝑥 times 𝑥. The common 𝑥 factors cancel, which
leaves us with the simplified expression 𝑥 plus 53 over 𝑥. But this simplified expression is
not equivalent to the 𝑛 sub one expression. So even though the domains were
equal, we still need to eliminate option (D).
Finally, we consider option
(E). We find that the denominator of 𝑥
squared plus 53𝑥 has two real number zeros: negative 53 and zero. These two values are excluded from
the domain, which makes it more restrictive than the domain of 𝑛 sub one. We can show that the expression
simplifies to one over 𝑥. But the domains are not equal, so
we eliminate this option anyways.
In conclusion, the two functions
given in option (C) are equal, based on the fact that they have the same domain and
simplify to the same expression.
