Question
Describe the domain and range of 𝑓(𝑥) = −2𝑥^2 and the domain and range of its
inverse.
inverse.
Answers
GPT 3.5
For the function 𝑓(𝑥) = −2𝑥^2, the domain is all real numbers because any real number can be squared and multiplied by -2. However, the range of this function is all real numbers less than or equal to 0, as squaring any real number results in a non-negative value, and multiplying it by -2 makes it negative or equal to zero.
The inverse of this function, denoted as 𝑓^(-1)(𝑥), can be found by switching the x and y variables and solving for the new y. So we have:
𝑥 = -2𝑦^2
𝑥/(-2) = 𝑦^2
√(-𝑥/2) = 𝑦
Therefore, the inverse function is 𝑓^(-1)(𝑥) = √(-𝑥/2). The domain of the inverse function is all real numbers less than or equal to 0, as we are taking the square root of a negative number for positive outputs. The range of the inverse function is all real numbers, because taking the square root of any real number yields a real number.
The inverse of this function, denoted as 𝑓^(-1)(𝑥), can be found by switching the x and y variables and solving for the new y. So we have:
𝑥 = -2𝑦^2
𝑥/(-2) = 𝑦^2
√(-𝑥/2) = 𝑦
Therefore, the inverse function is 𝑓^(-1)(𝑥) = √(-𝑥/2). The domain of the inverse function is all real numbers less than or equal to 0, as we are taking the square root of a negative number for positive outputs. The range of the inverse function is all real numbers, because taking the square root of any real number yields a real number.