Questions: Consider the function f(x)=3x^2-6x-3. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range.

Consider the function f(x)=3x^2-6x-3.
a. Determine, without graphing, whether the function has a minimum value or a maximum value.
b. Find the minimum or maximum value and determine where it occurs.
c. Identify the function's domain and its range.

Transcript text: Consider the function $f(x)=3 x^{2}-6 x-3$. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range.

Solution

Solution Steps

Step 1: Determining the Minimum/Maximum Value

The given quadratic function is $f(x) = 3x^2 - 6x - 3$. Since the coefficient of $x^2$ is 3, the parabola opens upwards, indicating a minimum value.

Step 2: Finding the Minimum/Maximum Value and Its Location

The vertex of the parabola, where the minimum value occurs, is at $x = -\frac{-6}{2*3} = 1$. Plugging this value into the function gives $f(1) = -6$, which is the minimum value of the function.

Step 3: Identifying the Domain and Range

The domain of any quadratic function is all real numbers, (-∞, +∞). The range, depending on whether the parabola opens upwards or downwards, is [-6, +∞).