Range Of F(x) = 2x² + 2: A Step-by-Step Solution

Hey guys! Today, we're diving deep into the fascinating world of functions, specifically focusing on how to determine the range of a quadratic function over a given interval. This is a crucial concept in mathematics, and mastering it will not only help you ace your exams but also give you a solid foundation for more advanced topics. We'll break down a specific example, the function f(x) = 2x² + 2, and explore how to find its range over the interval -2 ≤ x < 5. Let's get started!

What is the Range of a Function?

Before we tackle the problem at hand, let's quickly recap what the range of a function actually means. In simple terms, the range is the set of all possible output values (y-values) that a function can produce. Think of it like this: you plug in various x-values (the input), and the function spits out corresponding y-values (the output). The range is the collection of all those y-values. To effectively determine the range, it's important to understand how the function behaves and what its key characteristics are. For quadratic functions, this often involves looking at the vertex and the direction the parabola opens.

For example, consider the basic quadratic function f(x) = x². The smallest value this function can produce is 0 (when x = 0), and it can produce infinitely large positive values. Therefore, the range of f(x) = x² is all non-negative real numbers, often written as [0, ∞). Now, let's see how this concept applies to our specific function and interval.

Analyzing the Function f(x) = 2x² + 2

Our mission is to find the range of the function f(x) = 2x² + 2 over the interval -2 ≤ x < 5. This is a quadratic function, which means its graph is a parabola. Understanding the parabola's shape is key to finding the range. The coefficient of the x² term (which is 2 in our case) tells us that the parabola opens upwards because it's positive. This means the parabola has a minimum point, which is its vertex. The vertex is a crucial point because it represents the lowest (or highest, if the parabola opens downwards) value of the function. For our function, the vertex will give us the minimum value in the range.

To find the vertex, we can use the vertex form of a quadratic equation, which is f(x) = a(x - h)² + k, where (h, k) is the vertex. In our case, we can rewrite f(x) = 2x² + 2 as f(x) = 2(x - 0)² + 2. This tells us that the vertex is at the point (0, 2). So, the minimum value of the function is 2, which occurs when x = 0. This is a significant piece of information for determining our range. Remember, we're dealing with an interval, so we also need to consider the endpoints of the interval and how they affect the function's output.

Considering the Interval -2 ≤ x < 5

Now that we know the vertex and the minimum value of the function, we need to consider the interval -2 ≤ x < 5. This means we are only interested in the function's behavior for x-values between -2 (inclusive) and 5 (exclusive). The interval's endpoints are essential because they might give us the boundaries of our range. Since our parabola opens upwards, the function's values will increase as we move away from the vertex (x = 0) in either direction. Therefore, we need to evaluate the function at the endpoints of the interval to see what the maximum value might be.

Let's start with x = -2. Plugging this into our function, we get f(-2) = 2(-2)² + 2 = 2(4) + 2 = 8 + 2 = 10. So, at x = -2, the function value is 10. Now let's consider x = 5. Plugging this into our function, we get f(5) = 2(5)² + 2 = 2(25) + 2 = 50 + 2 = 52. However, it's crucial to note that our interval is -2 ≤ x < 5, which means x = 5 is not included. The function will approach 52 as x gets closer to 5, but it will never actually reach 52. This detail is vital for expressing the range correctly. We know that the function's minimum value is 2 (at the vertex) and its values increase as we move away from the vertex. Now we have all the pieces to determine the range.

Determining the Range: Putting it All Together

We've done the groundwork, guys! Now, let's piece together the information we've gathered to determine the range of f(x) = 2x² + 2 over the interval -2 ≤ x < 5. We know the function has a minimum value of 2 at the vertex (x = 0). We also know that f(-2) = 10 and that the function approaches 52 as x approaches 5, but never actually reaches it. Therefore, the range of the function over this interval includes all values from 2 (inclusive) up to, but not including, 52. This can be expressed mathematically as 2 ≤ f(x) < 52.

It’s important to consider both the minimum value (the vertex) and the behavior at the endpoints of the interval. The fact that the interval includes -2 but excludes 5 makes a difference in how we express the range. If the interval had included 5, the range would have been 2 ≤ f(x) ≤ 52. However, because 5 is excluded, we use the “less than” symbol (<) to indicate that 52 is not part of the range. This attention to detail is crucial for correctly answering questions about ranges and intervals. We found the answer!

Common Mistakes to Avoid

When finding the range of a function, especially a quadratic function over an interval, there are a few common pitfalls you want to avoid. One mistake is simply plugging in the endpoints of the interval and assuming those values represent the entire range. As we saw in our example, the vertex plays a critical role in determining the range, especially if it falls within the given interval. If you only looked at f(-2) and a hypothetical f(5) (had it been included), you might miss the minimum value of 2, which is a crucial part of the range.

Another common mistake is not paying close attention to whether the endpoints of the interval are included or excluded. This affects whether you use “≤” or “<” when expressing the range. Forgetting to consider the shape of the parabola (whether it opens upwards or downwards) is also a big no-no. This determines whether the vertex represents a minimum or maximum value, which is essential for finding the range. So, remember to always consider the vertex, the endpoints, and the direction the parabola opens to accurately determine the range.

Practice Makes Perfect

Finding the range of a function might seem tricky at first, but with practice, it becomes much easier. The key is to understand the underlying concepts and apply them systematically. Remember to identify the type of function, find its key features (like the vertex for quadratic functions), consider the given interval, and evaluate the function at relevant points. Don't be afraid to sketch a graph of the function; this can often provide valuable insights and help you visualize the range.

The more you practice, the more comfortable you'll become with these techniques, and the faster you'll be able to solve these types of problems. So, grab some practice questions, work through them step-by-step, and don't get discouraged if you make mistakes along the way. Mistakes are learning opportunities! Keep practicing, and you'll be a range-finding pro in no time. You got this, guys!

Conclusion

In conclusion, determining the range of a function, especially a quadratic function like f(x) = 2x² + 2, requires a solid understanding of the function's properties and how they interact with the given interval. By carefully considering the vertex, the endpoints of the interval, and the direction the parabola opens, we can accurately identify the set of all possible output values. Remember to pay attention to details like whether the endpoints are included or excluded, and practice regularly to build your skills.

We've covered a lot of ground today, from the basic definition of the range to common mistakes to avoid. I hope this guide has been helpful in demystifying the process of finding the range of a quadratic function. Keep exploring, keep learning, and most importantly, keep practicing. You're on your way to mastering this important mathematical concept. Until next time, happy problem-solving!