Video Transcript
What are the values of π₯ for which
the functions π of π₯ is equal to π₯ minus five and π of π₯ is equal to π₯ squared
plus two π₯ minus 48 are both positive?
Letβs begin by considering the
function π of π₯ is equal to π₯ minus five. If we want this to be positive, π
of π₯ must be greater than zero. This gives us π₯ minus five is
greater than zero. Adding five to both sides of this
inequality gives us π₯ is greater than five. π of π₯ is therefore positive on
the open interval five to β. It is a positive function for any
value greater than five. We will now repeat this process for
π of π₯. This gives us π₯ squared plus two
π₯ minus 48 is greater than zero. To solve any quadratic inequality
of this form, we firstly need to find the zeros by setting our function equal to
zero. π₯ squared plus two π₯ minus 48
equals zero.
This can be factored or factorized
into two sets of parentheses or brackets. The first term in each bracket is
π₯. The second terms need to have a
product of negative 48 and a sum of two. Six multiplied by eight is equal to
48. This means that negative six
multiplied by eight is equal to negative 48. Negative six plus eight is equal to
two. Our two sets of parentheses are π₯
minus six and π₯ plus eight. As the product of these two terms
is equal to zero, either π₯ minus six equals zero or π₯ plus eight is equal to
zero. Adding six to both sides of the
first equation gives us π₯ is equal to six. And subtracting eight from both
sides of the second equation gives us π₯ is equal to negative eight. This means that the function π of
π₯ is equal to zero when π₯ equals six and π₯ equals negative eight.
As our function is quadratic and
the coefficient of π₯ squared is positive, the graph will be u-shaped. This means that it is positive on
two sections, when π₯ is greater than six and when π₯ is less than negative
eight. The solution to the inequality π₯
squared plus two π₯ minus 48 is greater than nought is π₯ is less than negative
eight or π₯ is greater than six. This can also be written using
interval notation. π of π₯ is positive in the open
interval negative β to negative eight or the open interval six to β.
We want to work out the values of
π₯ where both functions are positive. Letβs consider a number line with
the key values five, negative eight, and six marked on. We know that π of π₯ is positive
for all values greater than five. π of π₯ is positive for all values
less than negative eight and greater than six. This means that both functions are
positive when π₯ is greater than six. This could also be written using
interval notation as the open interval six to β.
