Question Video: Determining Parity of Functions Mathematics

What are the values of π‘₯ for which the functions 𝑓(π‘₯) = π‘₯ βˆ’ 5 and 𝑔(π‘₯) = π‘₯Β² + 2π‘₯ βˆ’ 48 are both positive?

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Video Transcript

What are the values of π‘₯ for which the functions 𝑓 of π‘₯ is equal to π‘₯ minus five and 𝑔 of π‘₯ is equal to π‘₯ squared plus two π‘₯ minus 48 are both positive?

Let’s begin by considering the function 𝑓 of π‘₯ is equal to π‘₯ minus five. If we want this to be positive, 𝑓 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. 𝑓 of π‘₯ is therefore positive on the open interval five to ∞. It is a positive function for any value greater than five. We will now repeat this process for 𝑔 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 𝑔 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. 𝑔 of π‘₯ is positive in the open interval negative ∞ to negative eight or the open interval six to ∞.

We want to work out the values of π‘₯ where both functions are positive. Let’s consider a number line with the key values five, negative eight, and six marked on. We know that 𝑓 of π‘₯ is positive for all values greater than five. 𝑔 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 ∞.

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