a company has determined that its weekly profit is a function of the number of items that it sells. which…

a company has determined that its weekly profit is a function of the number of items that it sells. which equation could represent the weekly profit in thousands of dollars, y, when the company sells x items?\n$y^{2}=4x^{2}-100$\n$y=-x^{2}+50x - 300$\n$x=-y^{2}+60y - 400$\n$x^{2}=-6y^{2}+200$

a company has determined that its weekly profit is a function of the number of items that it sells. which equation could represent the weekly profit in thousands of dollars, y, when the company sells x items?\n$y^{2}=4x^{2}-100$\n$y=-x^{2}+50x - 300$\n$x=-y^{2}+60y - 400$\n$x^{2}=-6y^{2}+200$

Answer

Explanation:

Step1: Recall function definition

A function has one - output (y - value) for each input (x - value). For an equation to represent a function of x, when we solve for y, there should be at most one y for each x.

Step2: Analyze each option

  • For $y^{2}=4x^{2}-100$, solving for y gives $y = \pm\sqrt{4x^{2}-100}$, so there are two y - values for most x - values (when $4x^{2}-100\geq0$), not a function.
  • For $y=-x^{2}+50x - 300$, for each value of x, we can calculate exactly one value of y using the formula. This is a quadratic function of x.
  • For $x=-y^{2}+60y - 400$, solving for y would give a quadratic in y and for some x - values, there would be two y - values, not a function of x.
  • For $x^{2}=-6y^{2}+200$, solving for y gives $y=\pm\sqrt{\frac{200 - x^{2}}{6}}$ (when $200 - x^{2}\geq0$), not a function of x.

Answer:

$y=-x^{2}+50x - 300$