Video Transcript
If the domain of the function π of
π₯ equals negative five over π₯ squared minus eight π₯ plus π is the set of real
numbers minus the set containing four, determine the value of π.
To answer this question, letβs
begin by inspecting the function π of π₯. π of π₯ is actually a rational
function. Assuming π is a constant, itβs 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
π₯ that make the denominator equal to zero. Comparing this to the domain of π
of π₯, the set of real numbers minus the set containing the element four, we can
deduce that this element four, this value of π₯, must make the denominator zero. In other words, if π₯ is equal to
four, π₯ squared minus eight π₯ plus π must be equal to zero.
With this in mind, we can
substitute π₯ equals four into this expression, set it equal to zero, and solve for
π. Substituting π₯ equals four, and we
get four squared minus eight times four plus π equals zero. Thatβs 16 minus 32 plus π equals
zero or negative 16 plus π equals zero. To solve for π, we need to add 16
to both sides of our equation. And when we do, we find that π is
equal to 16. And so, given information about the
domain of π of π₯, we can determine the value of π is equal to 16.
Now, in fact, by substituting π
equals 16 back into the expression for the denominator, we can check our answer. Letβs set it equal to zero. So π₯ squared minus eight π₯ plus
16 equals zero. To solve this equation for π₯ and
to find any values of π₯ that we have to disregard from the domain of our function,
we factor the left-hand side. So π₯ minus four times π₯ minus
four equals zero. And this must mean there is only
one solution. Itβs the solution to the equation
π₯ minus four equals zero, which is π₯ equals four. We said this was the value of π₯
weβd have to disregard from the domain of π of π₯. So weβve satisfied our
criteria. And so π is equal to 16.
