Find the domain of \( q(x)=-2 x^{2}+3 x-5 \) Write your answer in interval notation. Question Help: \( \square \) Message instructor Submit Question

To determine the **domain** of the function \( q(x) = -2x^{2} + 3x - 5 \), let's analyze the equation step by step. ### Understanding the Function 1. **Type of Function**: \( q(x) = -2x^{2} + 3x - 5 \) is a **quadratic function** because the highest power of \( x \) is 2. 2. **Polynomial Functions**: Quadratic functions are a type of polynomial function. In general, polynomial functions are defined for all real numbers because you can plug any real number into the polynomial without causing any mathematical issues (like division by zero or taking the square root of a negative number). ### Determining the Domain - **Domain Definition**: The domain of a function is the set of all possible input values (usually \( x \)-values) for which the function is defined. - **For Quadratic Functions**: Since there are no restrictions (like square roots or denominators that could be zero) in \( q(x) = -2x^{2} + 3x - 5 \), the function is defined for all real numbers. ### Writing the Domain in Interval Notation - **Interval Notation for All Real Numbers**: When a function is defined for all real numbers, its domain in interval notation is written as: \[ (-\infty, \infty) \] ### Conclusion The domain of \( q(x) = -2x^{2} + 3x - 5 \) is all real numbers. **Answer:** \[ (-\infty, \infty) \]