Graphing And Determining The Domain And Range Of F(x) = 3x - 7

Hey guys! Today, we're diving into the exciting world of functions, specifically focusing on how to graph them and figure out their domain and range. We'll be tackling the function f(x) = 3x - 7. Don't worry, it's not as intimidating as it looks! We'll break it down step by step, making sure everyone understands the process. So, buckle up and let's get started!

Understanding the Function f(x) = 3x - 7

Our mission today is to graph the function f(x) = 3x - 7 and then determine its domain and range. But before we jump into graphing, let's understand what this function actually represents. The function f(x) = 3x - 7 is a linear function. Linear functions are those that, when graphed, produce a straight line. The general form of a linear function is f(x) = mx + b, where m represents the slope and b represents the y-intercept. Now, looking at our function, we can see that m = 3 and b = -7. This means that for every increase of 1 in x, the value of f(x) increases by 3, and the line crosses the y-axis at the point (0, -7).

The slope, represented by m, is a crucial element. It tells us how steep the line is. A positive slope, like our 3, means the line slopes upwards from left to right. A larger slope value indicates a steeper line. Think of it like climbing a hill; a steeper hill has a larger slope. The y-intercept, b, is where the line intersects the vertical y-axis. In our case, the line crosses the y-axis at -7. This gives us a starting point when we're graphing the function. Knowing the slope and y-intercept provides a solid foundation for visualizing the line's behavior. We can immediately picture a line that's sloping upwards quite steeply and crossing the y-axis at a negative value. This understanding helps us anticipate the graph's appearance and makes the graphing process much smoother. By recognizing this function as linear, we know we only need two points to draw its complete graph, a fact that will simplify our task significantly. Linear functions are fundamental in mathematics and have countless applications in real-world scenarios, from calculating distances and speeds to modeling financial growth.

Graphing the Function

Now, let's get to the fun part – graphing! To graph f(x) = 3x - 7, we need to find at least two points that lie on the line. Remember, a straight line is uniquely defined by two points, so once we have those, we can simply connect them to draw the graph. A simple and effective way to find these points is to choose some values for x and then calculate the corresponding values for f(x). Let's start with x = 0. When x = 0, f(0) = 3(0) - 7 = -7. So, the point (0, -7) is on the graph. This point is also the y-intercept, which we already knew from our earlier analysis of the function!

Next, let's choose another value for x. A good choice is x = 1. When x = 1, f(1) = 3(1) - 7 = -4. So, the point (1, -4) is also on the graph. We now have two points: (0, -7) and (1, -4). To graph the function, we plot these two points on a coordinate plane. The coordinate plane consists of a horizontal x-axis and a vertical y-axis. The point (0, -7) is located 0 units along the x-axis and -7 units along the y-axis. The point (1, -4) is located 1 unit along the x-axis and -4 units along the y-axis. Once we've plotted these points, we take a straightedge or ruler and draw a line that passes through both of them. Extend the line beyond the points on both ends to show that the function continues infinitely in both directions. This line represents the graph of the function f(x) = 3x - 7. Visually, you'll see a line that slopes upwards quite steeply as you move from left to right, confirming our understanding of the slope. Graphing is a powerful tool that allows us to visualize mathematical relationships and understand the behavior of functions at a glance.

Determining the Domain

The domain of a function refers to all the possible input values (x-values) for which the function is defined. In simpler terms, it's the set of all x-values that you can plug into the function and get a valid output. For linear functions like f(x) = 3x - 7, the domain is particularly straightforward. Linear functions are defined for all real numbers. This means you can plug in any real number for x, whether it's a positive number, a negative number, zero, a fraction, or an irrational number, and the function will produce a real number output. There are no restrictions on the x-values. We can express this mathematically by saying that the domain is the set of all real numbers, which is often denoted by the symbol ℝ. In interval notation, we write the domain as (-∞, ∞). This notation indicates that the domain extends infinitely in both the negative and positive directions. Visualizing the graph of a linear function also helps to confirm this. The line extends infinitely in both directions along the x-axis, meaning there are no breaks or gaps in the possible x-values. Understanding the domain is crucial because it tells us the boundaries of the function's input. It helps us avoid plugging in values that would lead to undefined results or errors. For linear functions, the domain is usually the entire set of real numbers, making them very versatile and applicable in a wide range of scenarios.

Determining the Range

Now that we've tackled the domain, let's move on to the range. The range of a function is the set of all possible output values (y-values) that the function can produce. It's essentially what you get out of the function after plugging in all the possible input values from the domain. Similar to the domain, the range of a linear function like f(x) = 3x - 7 is also all real numbers. This is because a non-horizontal linear function extends infinitely upwards and downwards. No matter what y-value you pick, you can always find an x-value that will produce that y-value. Think about it: as x gets very large in the positive direction, f(x) also gets very large in the positive direction. Conversely, as x gets very large in the negative direction, f(x) also gets very large in the negative direction. There are no horizontal asymptotes or other restrictions that would limit the possible output values.

We can express the range mathematically as the set of all real numbers, which, as we mentioned before, is denoted by the symbol ℝ. In interval notation, the range is also written as (-∞, ∞), indicating that it extends infinitely in both directions along the y-axis. Looking at the graph of f(x) = 3x - 7, you'll see the line stretching upwards and downwards without any breaks or gaps. This visual representation reinforces the idea that any y-value is attainable. The fact that both the domain and range of this linear function are all real numbers highlights a key characteristic of linear functions: they are continuous and unbounded. This means they have no breaks in their graph and can take on any value, both as inputs and outputs. Understanding the range is just as important as understanding the domain because it tells us the full spectrum of values that the function can generate.

Conclusion

Alright guys, we've successfully graphed the function f(x) = 3x - 7 and determined its domain and range! We learned that this function is a linear function with a slope of 3 and a y-intercept of -7. We found two points on the line, (0, -7) and (1, -4), plotted them on a coordinate plane, and drew a line through them to create the graph. We also discovered that the domain and range of this function are both all real numbers, which we expressed as ℝ or (-∞, ∞) in interval notation.

Understanding how to graph functions and determine their domain and range is a fundamental skill in mathematics. It allows us to visualize and analyze the behavior of functions, which is essential for solving problems in various fields, including science, engineering, and economics. Linear functions, in particular, are widely used in modeling real-world phenomena, making them a crucial concept to grasp. I hope this breakdown has been helpful and that you now feel more confident in your ability to tackle similar problems. Keep practicing, and you'll become a function graphing pro in no time!