Question

Determine the domain and range of the quadratic function. (Enter your answers using interval notation.) f(x) = −4(x + 1)2 − 4

Answer 1

Domain -∞ < x < +∞

Range f(x) ≤ - 4

Explanation:

The given quadratic function is f(x) = - 4 (x + 1)² - 4

Now, we have to find the domain and range of the quadratic function.

Now, it is clear from the given function that f(x) will have real value for all real value of x.

Therefore, the domain of the function is -∞ < x < +∞ (Answer)

Now, (x + 1)² is always ≥ 0

i.e. (x + 1)² ≥ 0

⇒ - 4 (x + 1)² ≤ 0 {Since the inequality sign changes for multiplying both sides of the equation by a negative term}

⇒ - 4 (x + 1)² - 4 ≤ - 4

⇒ f(x) ≤ - 4

Therefore, this is the required range of the function f(x). (Answer)