Questions: Find the domain and range of f(x)=-3x^2-6x-1 from its graph (shown below). Write your answers in interval notation.

Transcript text: Find the domain and range of $f(x)=-3 x^{2}-6 x-1$ from its graph (shown below). Write your answers in interval notation.

Solution

Solution Steps

Step 1: Identify the Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the quadratic function $ f(x) = -3x^2 - 6x - 1 $, there are no restrictions on the x-values. Therefore, the domain is all real numbers.

Step 2: Identify the Range

The range of a function is the set of all possible output values (y-values). Since the quadratic function $ f(x) = -3x^2 - 6x - 1 $ opens downwards (the coefficient of $ x^2 $ is negative), it has a maximum value at its vertex.

Step 3: Find the Vertex

To find the vertex of the quadratic function, use the vertex formula $ x = -\frac{b}{2a} $. Here, $ a = -3 $ and $ b = -6 $: \[ x = -\frac{-6}{2(-3)} = 1 \]

Substitute $ x = 1 $ back into the function to find the y-coordinate of the vertex: \[ f(1) = -3(1)^2 - 6(1) - 1 = -3 - 6 - 1 = -10 \]

So, the vertex is at $ (1, -10) $, and the maximum value of the function is -10.