For which values of 𝑥 is the
function 𝑛 of 𝑥 equals 𝑥 squared minus 25 over 𝑥 squared minus 12𝑥 plus 32 not
Let’s begin by inspecting the
function 𝑛 of 𝑥. 𝑛 of 𝑥 is the quotient of a pair
of polynomials. That is, it is a polynomial
function divided by a second polynomial function. In order to identify the values of
𝑥 for which the function is not defined, we’ll begin by considering the domain of a
rational function. The domain of a rational function,
of course, is the set of values of 𝑥 for which the function is defined. So, if we consider the set of
values of 𝑥 for which the function is defined, we’ll be able to quickly identify
the values for which it is not defined.
The domain of a rational function
is the set of real numbers, but we must exclude any values of 𝑥 that make the
denominator of that function equal to zero. This means that our function will
be defined over the set of real numbers excluding the set of numbers that make the
expression on the denominator, 𝑥 squared minus 12𝑥 plus 32, equal to zero.
To find such values of 𝑥, we will
set the denominator equal to zero and solve, that is, 𝑥 squared minus 12𝑥 plus 32
equals zero. Since we have a quadratic, we can
attempt to solve by first factoring this quadratic expression. We know that we must have an 𝑥 at
the beginning of each expression because 𝑥 times 𝑥 gives us the 𝑥 squared. Then, we need to find a pair of
numbers whose product is 32 and whose sum is negative 12. Well, negative four times negative
eight is positive 32 as required. But negative four plus negative
eight is indeed negative 12.
So we rewrite our equation as
shown. 𝑥 minus four times 𝑥 minus eight
is equal to zero. Then, of course, for the product of
these two expressions to be equal to zero, we know that either one or other of those
expressions must itself be zero. So the solutions to our equation
are given by the solutions to the equations 𝑥 minus four equals zero and 𝑥 minus
eight equals zero.
We solve our first equation by
adding four to both sides, so we get 𝑥 is equal to four. And we solve our second equation by
adding eight to both sides, so 𝑥 is equal to eight. Remember, if we’re thinking about
the domain of 𝑛 of 𝑥, we know it’s the set of real numbers minus the set
containing the numbers that make the denominator equal to zero. So the domain of our function is
the set of real numbers minus the set containing four and eight. Of course, this means our function
is defined over this set. And therefore, it must be undefined
when 𝑥 is equal to four or 𝑥 is equal to eight. And so we see that the function 𝑛
of 𝑥 is undefined for the set containing four and eight.