Question

What are the domain and range of the function f\left(x\right)=x^{4}-2x^{2}-4? domain: all real numbers range: all real numbers greater than or equal to –5 domain: all real numbers range: all real numbers greater than or equal to –4 domain: all real numbers between –2.5 and 2.5 range: all real numbers domain: all real numbers between –2.5 and 2.5 range: all real numbers greater than or equal to –5

Answer 1

Answer: domain: all real numbers; range: all real numbers greater than or equal to –4

Step-by-step explanation: The function f(x) = x^4 - 2x^2 - 4 is a polynomial function, which means its domain is all real numbers. To find the range of the function, we can complete the square for the quadratic term x^4 - 2x^2 to get f(x) = (x^2 - 1)^2 - 5. The minimum value of (x^2 - 1)^2 is 0, which occurs when x^2 = 1 or x = ±1. Therefore, the minimum value of f(x) is f(±1) = (±1)^4 - 2(±1)^2 - 4 = -5 + 1 = -4. Hence, the range of f(x) is all real numbers greater than or equal to -4.