If the domain of the function 𝑓 of
𝑥 equals negative five over 𝑥 squared minus eight 𝑥 plus 𝑘 is the set of real
numbers minus the set containing four, determine the value of 𝑘.
To answer this question, let’s
begin by inspecting the function 𝑓 of 𝑥. 𝑓 of 𝑥 is actually a rational
function. Assuming 𝑘 is a constant, it’s the
quotient of a pair of polynomials. We know that the domain of a
rational function is the set of all real numbers. But we must exclude any values of
𝑥 that make the denominator equal to zero. Comparing this to the domain of 𝑓
of 𝑥, the set of real numbers minus the set containing the element four, we can
deduce that this element four, this value of 𝑥, must make the denominator zero. In other words, if 𝑥 is equal to
four, 𝑥 squared minus eight 𝑥 plus 𝑘 must be equal to zero.
With this in mind, we can
substitute 𝑥 equals four into this expression, set it equal to zero, and solve for
𝑘. Substituting 𝑥 equals four, and we
get four squared minus eight times four plus 𝑘 equals zero. That’s 16 minus 32 plus 𝑘 equals
zero or negative 16 plus 𝑘 equals zero. To solve for 𝑘, we need to add 16
to both sides of our equation. And when we do, we find that 𝑘 is
equal to 16. And so, given information about the
domain of 𝑓 of 𝑥, we can determine the value of 𝑘 is equal to 16.
Now, in fact, by substituting 𝑘
equals 16 back into the expression for the denominator, we can check our answer. Let’s set it equal to zero. So 𝑥 squared minus eight 𝑥 plus
16 equals zero. To solve this equation for 𝑥 and
to find any values of 𝑥 that we have to disregard from the domain of our function,
we factor the left-hand side. So 𝑥 minus four times 𝑥 minus
four equals zero. And this must mean there is only
one solution. It’s the solution to the equation
𝑥 minus four equals zero, which is 𝑥 equals four. We said this was the value of 𝑥
we’d have to disregard from the domain of 𝑓 of 𝑥. So we’ve satisfied our
criteria. And so 𝑘 is equal to 16.