When Is A Linear Function Positive? Solving F(x) > 0

Jul 25, 2025 by ADMIN 53 views

Decoding Linear Functions When Does f(x) Turn Positive?

Hey there, math enthusiasts! Today, we're diving into the fascinating world of linear functions and figuring out when they turn positive. We've got a function, helpfully named f, that's both continuous and linear. This means it draws a straight line when you graph it, no breaks or sudden jumps. We're given a table of values, like a treasure map, showing us where the function has been. Our mission? To pinpoint the exact interval where f(x), our function's output, is happily above zero. So, let's put on our detective hats and solve this mathematical mystery, exploring the ins and outs of continuous linear functions and how to determine their positivity.

Unpacking the Linear Function Clues

First, let's break down what it means for f to be a continuous linear function. "Continuous" is math-speak for a smooth, unbroken line. You can trace it with your finger without lifting your pencil. "Linear" means it's a straight line, not a curve or squiggle. Think of it as the shortest distance between two points – that's the essence of a linear function. Now, let's look at the table of values. It's like a series of snapshots, showing us the function's height (f(x)) at different locations along the x-axis. We have these points:

When x is -3, f(x) is 4
When x is 0, f(x) is 3
When x is 3, f(x) is 0

Notice anything interesting? As x increases, f(x) seems to be decreasing. This is a crucial observation because it tells us our line is sloping downwards. This makes sense since linear functions have a constant rate of change. Now, the big question: where does this line cross the x-axis (where f(x) becomes zero) and, more importantly, where is it above the x-axis (where f(x) is positive)? We already know that at x=3, f(x)=0.

The Quest for Positivity

Our main keyword here is pinpointing the interval over which f(x) is positive. Remember, positive values of f(x) mean the function's graph is above the x-axis. Looking at our data points, we see that f(x) is 4 when x is -3 and 3 when x is 0. Both of these are positive! This tells us that, at least for these x-values, we're in the positive zone. We also know that at x=3, f(x) hits 0. Since it's a straight line, the function must be positive for all x-values less than 3.

But how far does this positivity extend? Since the function is decreasing, it will eventually cross the x-axis and become negative. We already know it crosses at x=3, so before that point, it's positive. Therefore, the interval where f(x) is positive is all x-values less than 3. In mathematical notation, we'd write this as x < 3. Or, in interval notation, we'd express it as (-∞, 3). So, we've cracked the code! By understanding the properties of continuous linear functions and carefully analyzing our data, we've successfully identified the interval where f(x) is positive.

Cracking the Code: Solving the Linear Function Problem

So, you've got a linear function puzzle on your hands, huh? No sweat! We're going to break down this problem step-by-step, turning those head-scratching moments into "Aha!" moments. The key here is understanding what a linear function really is. It's like a straight path, a line drawn on a graph, with no curves or sudden detours. And because it's a line, its behavior is super predictable. That's why, with just a few clues, we can figure out its entire story.

In this case, our clues are in the form of a table, which tells us the f(x) values at specific x points. This is like having a few snapshots of the line at different locations. We can see where it is, and, more importantly, we can see the trend: Is it going up? Is it going down? Is it staying flat? This trend, this direction, is what will ultimately lead us to the answer. The question is asking us to pinpoint the interval where the function's value, f(x), is positive. Remember, positive values are above the x-axis. So, we're essentially looking for the section of our line that's hanging out in the sunny, upper part of the graph.

Now, let's dig into the actual process of finding that interval. We'll start by carefully examining the table, looking for patterns and key points. We're on the hunt for where the function crosses that x-axis, where it transitions from positive to negative (or vice-versa). This is our turning point, our landmark. Once we find that landmark, we can use the linear nature of the function to extrapolate. Because it's a straight line, the behavior on either side of that landmark is consistent and predictable. So, let's roll up our sleeves, dive into those numbers, and solve this puzzle together!

Step-by-Step Solution: Finding the Positive Territory

Alright, let's get down to business and work through this linear function problem. Our goal, remember, is to identify the interval where f(x) is positive. Let's start by re-examining the provided table. This table is our treasure map, guiding us to the solution. Each pair of x and f(x) values is like a breadcrumb, leading us along the path of the function. We've got the following pairs:

x = -3, f(x) = 4
x = 0, f(x) = 3
x = 3, f(x) = 0

The first thing that jumps out is that f(x) is decreasing as x increases. This is a crucial observation! It tells us that our line is sloping downwards. Picture it in your mind: a line starting high up and gradually descending as it moves to the right. This is a key piece of the puzzle. We also see that f(x) is 0 when x is 3. This is another vital piece of information. It's the point where our line crosses the x-axis, the boundary between the positive and negative f(x) values. Think of it as a gate. On one side of the gate, f(x) is positive; on the other side, it's negative. Since our line is sloping downwards, we know that it must be positive before it crosses the x-axis at x = 3. In other words, for all x values less than 3, f(x) is positive.

Now, let's put this in mathematical language. The interval where f(x) is positive is x < 3. We can also express this using interval notation: (-∞, 3). This notation simply means all numbers from negative infinity up to, but not including, 3. And there you have it! We've successfully navigated the problem, identified the critical points, and determined the interval where our linear function is happily in positive territory.

Putting It All Together: The Positive Interval

Let's recap what we've learned about figuring out when a linear function is positive. We started with the fundamental understanding of what a linear function is – a straight line. This straightness is key because it means the function's behavior is consistent and predictable. Then, we analyzed the table of values, treating it as our set of clues. We looked for trends, like whether the function was increasing or decreasing, and we identified key points, especially where the function crossed the x-axis. This crossing point is where the function's sign changes, marking the boundary between positive and negative values.

For this specific problem, we saw that the function f(x) was decreasing and crossed the x-axis at x = 3. This meant that for all x-values less than 3, the function was positive. We expressed this interval in two ways: x < 3 and (-∞, 3). Both notations convey the same information, just in slightly different formats. The x < 3 notation is a simple inequality, while the (-∞, 3) notation is interval notation, which is a concise way to represent a range of numbers. So, the takeaway here is that by combining our understanding of linear functions with careful analysis of the given data, we can confidently pinpoint the interval where the function is positive. And that's a valuable skill to have in your mathematical toolkit!

Answering the Question: The Interval of Positivity

So, let's get straight to the answer, guys! We've journeyed through the world of linear functions, navigated the ups and downs of f(x), and now we're ready to declare our findings. The question, in case you forgot, was: Over which interval of the domain is the function f positive? After our careful analysis of the table and our understanding of how linear functions behave, we've arrived at a clear and concise answer.

The interval where f(x) is positive is when x is less than 3. That's it! Simple and sweet. But let's not forget the journey we took to get here. We looked at the trend of the function, noticing it was decreasing. We identified the point where it crossed the x-axis. And we used this information to deduce that the function must be positive before that crossing point. Remember, understanding the process is just as important as getting the right answer. It's like knowing how to fish, rather than just being given a fish. So, next time you encounter a similar problem, you'll have the tools and the knowledge to tackle it with confidence. And that, my friends, is the real reward in mathematics. You can also represent this answer using different notations, such as x < 3 or (-∞, 3), but the core concept remains the same: the function f is positive for all values of x that are less than 3.

Final Thoughts on Linear Functions and Positivity

Alright, we've reached the end of our linear function adventure, guys! We've successfully pinpointed the interval where our function f(x) is positive, and hopefully, you've gained a deeper understanding of linear functions in the process. Remember, these functions are the straight shooters of the math world – predictable, consistent, and always following a straight line. This predictability is what makes them so useful and so easy to analyze. By understanding their fundamental properties, we can unlock their secrets and solve problems like the one we tackled today.

Finding the interval of positivity is a common task when working with functions, and it's a skill that translates to many areas of mathematics and beyond. Whether you're analyzing data, modeling real-world phenomena, or simply exploring the beauty of mathematics, the ability to understand and interpret functions is crucial. So, keep practicing, keep exploring, and keep asking questions. The world of mathematics is vast and fascinating, and there's always something new to discover! And hey, you guys can surely go further than that! Next time, we can explore quadratic functions, or even more complex mathematical beasts. Remember, every mathematical challenge is just another opportunity to learn and grow. So, keep that curiosity burning, and who knows what mathematical adventures await you in the future!