Question

The graph of the function showing the path of a soccer ball thrown by a goalie in a video game, f(x) = –0.05x2 + 2.5x + 5.2, is used to determine the height of the ball depending on its distance from the goalie, x.

How do the mathematical domain and reasonable domain compare?

mathematical: 0 < x < ∞
reasonable: 0 ≤ x ≤ 25
mathematical: 0 < x < ∞
reasonable: 0 ≤ x ≤ 52
mathematical: −∞ < x < ∞
reasonable: 0 ≤ x ≤ 25
mathematical: −∞ < x < ∞
reasonable: 0 ≤ x ≤ 52

Answer 1

The correct option is 4.

Explanation:

The given function is

`f(x)=-0.05x^2+2.5x+5.2`

Where f(x) is height of the ball and x is the distance.

It is a polynomial function with degree 2. All polynomial functions are defined for all real numbers, therefore the mathematical domain of the function is all real numbers.

`-\infty<x<\infty`

Factorize the given function.

`f(x)=-0.05(x^2-50x-104)`

`f(x)=-0.05(x^2-50x-104)`

`f(x)=-0.05(x^2-52x+2x-104)`

`f(x)=-0.05(x(x-52)+2(x-52))`

`f(x) = -0.05 (x - 52) (x + 2)`

Put f(x)=0 to find the x intercepts.

`0 = -0.05 (x - 52) (x + 2)`

Equate each factor equal to 0.

`x=52,-2`

Therefore at x=52 and -2, the graph of f(x) intersects x-axis. Before x=-2 and after x=52 the values of f(x) is negative. Height cannot be negative, therefore reasonable domain is lie between -2 to 52.

Distance cannot be negative, therefore the reasonable domain must be positive.

`0\leq x\leq 52`

Therefore the reasonable domain is
`0\leq x\leq 52` and option 4 is correct.