Finding Solutions For F(x)=|2x-1|-3 When F(x)=0

by Luna Greco 48 views

Hey guys! Today, we're diving into a fun little math problem. We're going to find the values of $x$ that make the function $f(x)=|2x-1|-3$ equal to zero. This might sound intimidating at first, but trust me, it's totally doable! We'll break it down step by step, so you'll be a pro at solving these kinds of problems in no time. Get ready to put on your thinking caps, and let's get started!

Understanding Absolute Value Functions

Before we jump into solving $f(x) = 0$, let's quickly recap what absolute value functions are all about. Absolute value, at its core, is the distance a number is from zero. It doesn't care about direction, so whether you're dealing with a positive or negative number, the absolute value is always non-negative. Think of it as the magnitude of a number. Mathematically, we write the absolute value of $x$ as $|x|$. For example, $|5| = 5$ and $|-5| = 5$. See? The negative sign vanishes!

Now, when we throw in a function like $f(x) = |2x - 1| - 3$, things get a bit more interesting. The absolute value part, $|2x - 1|$, means we need to consider two scenarios: when the expression inside the absolute value is positive or zero, and when it's negative. This is because the absolute value "flips" the sign of any negative input to make it positive. This piecewise nature of absolute value functions is key to solving equations involving them. We're essentially dealing with two different linear equations disguised within a single absolute value expression. So, to conquer this, we need to treat each scenario separately, which we'll dive into in the next section. Remember, understanding the fundamental concept of absolute value – distance from zero – is crucial for tackling these problems. It's not just about blindly applying rules, but grasping the underlying idea that makes the whole process click.

Solving $f(x) = 0$: The Two Scenarios

Okay, let's get down to business and solve $f(x) = |2x - 1| - 3 = 0$. As we discussed, the absolute value throws a curveball, and we need to handle it by considering two distinct scenarios. This is where the magic happens, guys!

Scenario 1: When the expression inside the absolute value is non-negative, i.e., $2x - 1 ≥ 0$

In this case, the absolute value bars simply vanish, and we can treat the expression inside as is. So, our equation becomes:

2x−1−3=02x - 1 - 3 = 0

Let's simplify this bad boy:

2x−4=02x - 4 = 0

Now, we isolate $x$:

2x=42x = 4

Divide both sides by 2, and boom!

x=2x = 2

But hold on a second! We need to make sure this solution actually fits our initial condition, $2x - 1 ≥ 0$. Let's plug $x = 2$ back in:

2(2)−1=32(2) - 1 = 3

Since 3 is indeed greater than or equal to 0, our solution $x = 2$ is valid for this scenario. Phew!

Scenario 2: When the expression inside the absolute value is negative, i.e., $2x - 1 < 0$

Now, things get a tiny bit trickier. When $2x - 1$ is negative, the absolute value flips its sign. So, $|2x - 1|$ becomes $-(2x - 1)$. Our equation transforms into:

−(2x−1)−3=0-(2x - 1) - 3 = 0

Let's distribute the negative sign:

−2x+1−3=0-2x + 1 - 3 = 0

Simplify:

−2x−2=0-2x - 2 = 0

Isolate $x$:

−2x=2-2x = 2

Divide both sides by -2:

x=−1x = -1

Again, we need to check if this solution satisfies our condition, $2x - 1 < 0$. Let's plug in $x = -1$:

2(−1)−1=−32(-1) - 1 = -3

Since -3 is less than 0, our solution $x = -1$ is also valid! Woohoo!

So, after tackling both scenarios, we've found two values of $x$ that make $f(x) = 0$. That's how we conquer these absolute value equations, guys. Break them down, consider the scenarios, and always, always check your solutions!

Verifying the Solutions

Alright, we've found our potential solutions: $x = 2$ and $x = -1$. But, as any good mathematician knows, we're not done until we've verified our answers. This is a crucial step to make sure we haven't made any sneaky errors along the way. Plugging our solutions back into the original equation is like giving our answer a final exam – it has to pass to get the stamp of approval!

Let's start with $x = 2$. We'll substitute it into $f(x) = |2x - 1| - 3$:

f(2)=∣2(2)−1∣−3f(2) = |2(2) - 1| - 3

Simplify inside the absolute value:

f(2)=∣4−1∣−3f(2) = |4 - 1| - 3

f(2)=∣3∣−3f(2) = |3| - 3

The absolute value of 3 is simply 3:

f(2)=3−3f(2) = 3 - 3

And finally:

f(2)=0f(2) = 0

Awesome! $x = 2$ checks out. It makes the function equal to zero, just as we wanted.

Now, let's put $x = -1$ to the test:

f(−1)=∣2(−1)−1∣−3f(-1) = |2(-1) - 1| - 3

Simplify inside the absolute value:

f(−1)=∣−2−1∣−3f(-1) = |-2 - 1| - 3

f(−1)=∣−3∣−3f(-1) = |-3| - 3

The absolute value of -3 is 3:

f(−1)=3−3f(-1) = 3 - 3

And again:

f(−1)=0f(-1) = 0

Fantastic! $x = -1$ also passes the test. Both of our solutions hold up under scrutiny. This verification step not only confirms our answers but also gives us that extra bit of confidence that we've nailed the problem. Remember, guys, always verify your solutions, especially in math! It's like the cherry on top of a perfectly solved problem.

Graphical Interpretation

Let's take a moment to visualize what we've just solved. Sometimes, seeing the problem from a graphical perspective can give us a deeper understanding. We've found the values of $x$ where $f(x) = |2x - 1| - 3 = 0$. Graphically, this means we've found the points where the graph of the function $f(x)$ intersects the x-axis. These points are also known as the x-intercepts or the zeros of the function.

The graph of $f(x) = |2x - 1| - 3$ is a V-shaped graph due to the absolute value. The vertex of the V is where the expression inside the absolute value, $2x - 1$, equals zero. Solving $2x - 1 = 0$ gives us $x = 1/2$. This is the x-coordinate of the vertex. To find the y-coordinate, we plug $x = 1/2$ back into the function:

f(1/2)=∣2(1/2)−1∣−3=∣0∣−3=−3f(1/2) = |2(1/2) - 1| - 3 = |0| - 3 = -3

So, the vertex of the V is at the point $(1/2, -3)$. The graph opens upwards because the coefficient of the absolute value term is positive.

Now, we found that the function equals zero when $x = 2$ and $x = -1$. These are the x-intercepts of the graph. If you were to sketch the graph, you'd see that the V-shaped graph crosses the x-axis at these two points. The left side of the V crosses at $x = -1$, and the right side crosses at $x = 2$.

This graphical interpretation provides a visual confirmation of our algebraic solutions. It's like having a second opinion that agrees with our calculations! Understanding the graphical representation of functions is a powerful tool in mathematics. It allows us to connect the algebra we do with the visual picture, making the concepts more intuitive and easier to grasp. So, guys, next time you're solving an equation, try sketching a graph – it might just give you a whole new perspective!

Conclusion

Alright, guys! We've reached the end of our journey to find the zeros of the function $f(x) = |2x - 1| - 3$. We started by understanding the essence of absolute value functions – how they deal with positive and negative inputs. We then broke down the problem into two scenarios based on the expression inside the absolute value. We carefully solved each scenario, remembering to check if our solutions satisfied the initial conditions. After finding our potential answers, $x = 2$ and $x = -1$, we put them to the test by verifying them in the original equation. And finally, we explored the graphical interpretation of our solutions, seeing how they represent the x-intercepts of the function's graph.

This problem is a fantastic example of how breaking down a complex task into smaller, manageable steps can lead to a successful solution. The key takeaways here are:

  • Understand the basics: A solid grasp of absolute value is crucial.
  • Consider all scenarios: Absolute value equations often require splitting into cases.
  • Verify your solutions: Always double-check your answers to avoid errors.
  • Visualize the problem: Graphs can provide valuable insights.

Solving equations involving absolute values might seem daunting at first, but with practice and a systematic approach, you can conquer them like a math whiz! Keep practicing, keep exploring, and most importantly, keep having fun with math! Remember, guys, math is not just about numbers and equations; it's about problem-solving, logical thinking, and the thrill of discovering solutions. So, keep that curiosity alive, and you'll be amazed at what you can achieve!