Unity Games is a company that sells video games. Its revenue last year was modelled by the function g(x)=-4x^4+6x(4x-1)-4 and its revenue this year is modelled by the function f(x)=-3x^2((x^2)-8)-6x-5 where x is the number of video games sold in the thousands and the revenue is in millions of dollars. Determine the range of values for the number of video games sold that makes the revenue this year greater than the revenue last year. You must use a full algebraic solution.
so you want f(x) > g(x), right?
-4x^4+6x(4x-1)-4 > -3x^2((x^2)-8)-6x-5
That is,
-4x^4+6x(4x-1)-4 - (-3x^2(x^2-8)-6x-5) > 0
-x^4+1 = 0
-(x-1)(x+1)(x^2+1) > 0
-1 < x < 1
That is, fewer than 1000 games
To determine the range of values for the number of video games sold that makes the revenue this year greater than the revenue last year, we need to compare the two revenue functions and find the values of x for which f(x) is greater than g(x).
First, we'll set up the inequality:
f(x) > g(x)
Substituting the given functions:
-3x^2(x^2 - 8) - 6x - 5 > -4x^4 + 6x(4x - 1) - 4
Now, let's simplify the inequality:
-3x^4 + 24x^2 - 6x - 5 > -4x^4 + 24x^2 - 6x - 4
We can cancel out the common terms (-6x and -5):
-3x^4 > -4x^4 - 4
Next, we'll move all terms to one side of the inequality:
-x^4 > -4
Remember that we need to reverse the direction of the inequality when multiplying or dividing both sides by a negative number.
Dividing both sides by -1, we get:
x^4 < 4
Taking the fourth root of both sides, we get:
x < ∛4
Now, let's evaluate the right side:
x < 1.587
Finally, we need to convert the x-values from thousands to millions. Since the given functions have x in thousands, we need to divide the inequality bound by 1000:
x < 1.587 / 1000
Simplifying:
x < 0.001587
Therefore, the range of values for the number of video games sold that makes the revenue this year greater than the revenue last year is x < 0.001587 million games sold (or approximately x < 1587 games sold).