As an expert in the field of mathematics, I can guide you through the process of finding the range of a function. The range of a function is the set of all possible output values (y-values) that you can get from the function. To find the range, you typically need to analyze the function's formula and behavior. Here's a step-by-step guide on how to do this:

### Step 1: Understand the Function's Formula

The first step is to clearly understand the function's formula. Let's use the example you provided: f(x) = 3x^2 + 6x - 2. This is a quadratic function, which means it has a parabolic shape. Quadratic functions can have a minimum or maximum value, depending on whether they open upwards (minimum) or downwards (maximum).

### Step 2: Determine if the Function Has a Maximum or Minimum Value

For a quadratic function f(x) = ax^2 + bx + c, if a > 0, the parabola opens upwards, and the function has a minimum value. If a < 0, it opens downwards, and the function has a maximum value. In our example, a = 3, which is positive, so the function has a minimum value.

### Step 3: Find the Vertex of the Function

The vertex of a parabola is the point at which the function reaches its minimum or maximum value. For a quadratic function in the form of f(x) = ax^2 + bx + c, the x-coordinate of the vertex is found using the formula x = -b / (2a).

Using our example:
x = -6 / (2 * 3) = -1

Now, you can find the y-coordinate by plugging this x-value back into the function:
**f(-1) = 3(-1)^2 + 6(-1) - 2 = 3 - 6 - 2 = -5**

So, the vertex is at the point (-1, -5).

### Step 4: Analyze the Behavior of the Function

Since our function has a minimum value, as x approaches positive or negative infinity, the value of f(x) will increase without bound. This means the range of the function is all real numbers greater than or equal to the y-value of the vertex.

### Step 5: Write Down the Range

For our function, the range is:
y ≥ -5

This means that the function can take any value from -5 to infinity.

### Step 6: Graph the Function (Optional)

Graphing the function can help visualize the behavior and confirm the range. You can use graphing tools or software to plot the function and see the shape of the parabola.

### Conclusion

Finding the range of a function involves understanding its formula, determining the type of behavior (minimum or maximum), finding the vertex if applicable, and analyzing the function's behavior as x approaches extreme values. For a quadratic function like the one we've been discussing, the range is determined by the direction of the parabola and the location of its vertex.

Now, let's move on to the next step.

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Method 1 Finding the Range of a Function Given a FormulaWrite down the formula. Let's say the formula you're working with is the following: f(x) = 3x2 + 6x -2. ... Find the vertex of the function if it's quadratic. ... Find a few other points in the function. ... Find the range on the graph.read more >>