Find the domain of $r(x)=-4 x^2+2 x+8$.
Write your answer in interval notation.
To find the domain of the function \(r(x) = -4x^2 + 2x + 8\), we first examine its structure. The function is a quadratic function, which is a specific type of polynomial function. A polynomial is an expression consisting of variables, coefficients, and non-negative integer exponents, and it involves only the operations of addition, subtraction, and multiplication.
Next, consider the constraints on the domain of a function. The domain is defined as the set of all possible input values (x-values) for which the function is defined and produces a real number as an output. For polynomial functions, there are no operations such as division by a variable or taking the square root of a variable that could potentially lead to undefined values (like division by zero or square roots of negative numbers). Since a polynomial is defined for all real numbers \(x\), we can substitute any real value into the function and obtain a valid result.
Because there are no values of \(x\) that make the expression \(r(x) = -4x^2 + 2x + 8\) undefined, the function is defined for all real numbers. In interval notation, the set of all real numbers is expressed as \((-\infty, \infty)\). Therefore, the domain of the function \(r(x)\) is the interval \((-\infty, \infty)\).
The concept of domain is crucial in real-world modeling. For instance, if you were modeling the trajectory of a projectile thrown into the air, the function \(h(t) = -16t^2 + v_0t + h_0\) represents its height at time \(t\). While the mathematical domain of this quadratic function is all real numbers, the practical domain is limited by the time the object is in flight, starting from \(t=0\) and ending when it hits the ground. Understanding the mathematical domain is the first step in identifying the context-specific interval of validity for such physical models.