Domain And Range Of F(x) = 2x² + 2x + 1 Explained
Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of functions, specifically the quadratic function f(x) = 2x² + 2x + 1. Our mission? To unravel its domain and range. These two concepts are fundamental to understanding any function, as they define the set of possible input values (domain) and the resulting output values (range). So, buckle up and let's embark on this mathematical adventure together!
Understanding Domain and Range: The Foundation of Functions
Before we jump into the specifics of our function, let's take a moment to solidify our understanding of domain and range. Think of a function as a machine: you feed it an input, and it spits out an output. The domain is the collection of all possible inputs you can feed into the machine without causing it to break down (i.e., produce an undefined result). The range, on the other hand, is the set of all possible outputs that the machine can produce.
For example, consider a simple function like f(x) = 1/x. You can plug in any number for x except for 0, because division by zero is undefined. Therefore, the domain of this function is all real numbers except 0. The range is also all real numbers except 0, because no matter what value you plug in for x, you'll never get an output of 0.
In the context of functions, the domain represents the set of all permissible x-values, while the range encompasses all the resulting y-values. Visualizing a function's graph can be incredibly helpful in determining its domain and range. The domain corresponds to the extent of the graph along the x-axis, while the range corresponds to the extent along the y-axis. Understanding these concepts is crucial for effectively analyzing and interpreting functions across various mathematical disciplines.
Delving into Our Quadratic Function: f(x) = 2x² + 2x + 1
Now that we've refreshed our understanding of domain and range, let's turn our attention to the star of the show: the quadratic function f(x) = 2x² + 2x + 1. This is a special type of function, characterized by its highest power of x being 2. Quadratic functions have a distinctive U-shaped graph called a parabola, which opens either upwards or downwards depending on the coefficient of the x² term. In our case, the coefficient is 2, which is positive, so the parabola opens upwards.
Quadratic functions are ubiquitous in mathematics and the real world. They model a wide range of phenomena, from the trajectory of a projectile to the shape of a suspension bridge. Their unique properties, including their vertex (the minimum or maximum point) and axis of symmetry, make them powerful tools for problem-solving and analysis. To effectively determine the domain and range of a quadratic function, it's essential to understand its graphical behavior and key features.
Determining the Domain of f(x) = 2x² + 2x + 1: No Restrictions Here!
The domain question is often the easier one to tackle, especially for polynomial functions like our quadratic. Are there any values of x that we can't plug into the function f(x) = 2x² + 2x + 1? Can we square any number? Yes. Can we multiply any number by 2? Absolutely. Can we add any numbers together? You bet!
There are no restrictions on the values of x that we can input into this function. We can square any real number, multiply it by 2, add 2 times the original number, and add 1, and we'll always get a real number as a result. This means that the domain of our function is all real numbers. We can express this mathematically using interval notation as (-∞, ∞). This notation signifies that the domain extends infinitely in both the negative and positive directions along the number line.
So, when it comes to the domain of quadratic functions, you can generally breathe a sigh of relief. Unless there are specific constraints imposed by the context of the problem (like a physical limitation), the domain will almost always be all real numbers. This is a characteristic feature of polynomials, which are well-behaved functions that don't have the types of restrictions you might encounter with functions involving fractions or square roots.
Unveiling the Range of f(x) = 2x² + 2x + 1: Finding the Minimum
Now for the trickier part: the range. Since our parabola opens upwards, it has a minimum point, also known as the vertex. The y-coordinate of this vertex will be the lowest value in the range. To find the vertex, we can use a couple of methods.
Method 1: Completing the Square
Completing the square is a powerful algebraic technique that allows us to rewrite the quadratic expression in a form that reveals the vertex. Let's do it!
f(x) = 2x² + 2x + 1
First, factor out the coefficient of the x² term (which is 2) from the first two terms:
f(x) = 2(x² + x) + 1
Now, we need to add and subtract a term inside the parentheses to complete the square. To find this term, we take half of the coefficient of the x term (which is 1), square it (which gives us 1/4), and add and subtract it inside the parentheses:
f(x) = 2(x² + x + 1/4 - 1/4) + 1
Next, we rewrite the expression inside the parentheses as a perfect square:
f(x) = 2((x + 1/2)² - 1/4) + 1
Distribute the 2:
f(x) = 2(x + 1/2)² - 1/2 + 1
Finally, simplify:
f(x) = 2(x + 1/2)² + 1/2
Now we can clearly see the vertex form of the quadratic! The vertex is at the point (-1/2, 1/2). Since the parabola opens upwards, the minimum value of the function is 1/2.
Method 2: Using the Vertex Formula
There's also a handy formula to directly calculate the x-coordinate of the vertex: x = -b / 2a, where a and b are the coefficients of the x² and x terms, respectively. In our case, a = 2 and b = 2, so:
x = -2 / (2 * 2) = -1/2
To find the y-coordinate of the vertex, we plug this x-value back into the original function:
f(-1/2) = 2(-1/2)² + 2(-1/2) + 1 = 1/2 - 1 + 1 = 1/2
Again, we find that the vertex is at (-1/2, 1/2), and the minimum value of the function is 1/2.
Expressing the Range: It's All About the Minimum
Since the parabola opens upwards and the minimum value is 1/2, the range of the function is all real numbers greater than or equal to 1/2. In interval notation, we write this as [1/2, ∞). The square bracket indicates that 1/2 is included in the range, while the parenthesis indicates that infinity is not a specific number but rather a concept of unboundedness.
Therefore, the range of f(x) = 2x² + 2x + 1 consists of all y-values that are greater than or equal to the y-coordinate of the vertex. This makes intuitive sense when you visualize the parabola: it extends upwards indefinitely, but it has a lowest point at the vertex.
The Grand Finale: Domain and Range Unveiled!
So, after our mathematical exploration, we've successfully determined the domain and range of the function f(x) = 2x² + 2x + 1. The domain, as we established, is all real numbers, or (-∞, ∞). The range, on the other hand, is all real numbers greater than or equal to 1/2, or [1/2, ∞). Therefore, the correct answer is:
B. The domain is (-∞, ∞). The range is [0.5, ∞).
Congratulations, math detectives! We've cracked the case of the domain and range for this quadratic function. Remember, understanding these concepts is crucial for a deeper understanding of functions and their applications in various fields.
Final Thoughts: Mastering Domain and Range
Mastering the concepts of domain and range is like gaining a superpower in the world of mathematics. It allows you to analyze and interpret functions with confidence, predict their behavior, and solve a wide range of problems. While quadratic functions are a great starting point, the principles we've discussed today apply to all types of functions, from simple linear functions to more complex trigonometric and exponential functions.
So, keep practicing, keep exploring, and never stop questioning. The world of mathematics is full of fascinating mysteries waiting to be unraveled, and understanding domain and range is a key step on your journey to mathematical mastery. Keep up the great work, guys!