Finding The Zero Of A Function F(x) = -2x + 5 A Step-by-Step Solution

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Hey guys! Ever found yourself staring at a function and wondering where it crosses the x-axis? That magical point is called the zero of the function, and it's super important in math and real-world applications. Today, we're going to dive deep into how to find the zero of a function, using the example f(x) = -2x + 5. We'll break it down step by step, so even if you're just starting with algebra, you'll get the hang of it. So, buckle up, and let's get started!

Understanding the Zero of a Function

Before we jump into solving, let's make sure we're all on the same page about what the zero of a function actually means. In simple terms, the zero of a function is the value of x that makes the function equal to zero. Graphically, it's the point where the function's line or curve intersects the x-axis. This is a fundamental concept in algebra and calculus, and it has wide-ranging applications in various fields, including physics, engineering, and economics.

Why Finding Zeros Matters

Finding the zeros of a function isn't just an abstract mathematical exercise; it has practical implications. For instance, in physics, the zeros of a function might represent the points where an object's trajectory intersects the ground. In engineering, they could represent the equilibrium points of a system. In economics, they might indicate the break-even points where costs equal revenue. Understanding how to find these zeros allows us to solve real-world problems and make informed decisions. Let's consider a few examples to illustrate this further:

  • Physics: Imagine you're analyzing the trajectory of a projectile, like a ball thrown into the air. The function describing the height of the ball over time might be a quadratic equation. The zeros of this function would represent the times when the ball hits the ground. Knowing these times can help you calculate the range of the projectile or optimize its launch angle.
  • Engineering: In circuit analysis, you might encounter functions that describe the voltage or current in a circuit over time. The zeros of these functions could represent the moments when the voltage or current is zero, which can be crucial for understanding the behavior of the circuit and preventing damage to its components.
  • Economics: In business, you might use functions to model the cost and revenue of a product. The zeros of the function representing the difference between revenue and cost (i.e., the profit function) would represent the break-even points where the business neither makes a profit nor incurs a loss. This information is essential for making pricing and production decisions.

As you can see, finding the zeros of a function is a versatile skill that can be applied in many different contexts. Now that we understand the importance of zeros, let's get back to our specific function, f(x) = -2x + 5, and find its zero.

Solving for the Zero of f(x) = -2x + 5

Okay, let's get down to business and find the value of x that makes our function, f(x) = -2x + 5, equal to zero. The key here is to set the function equal to zero and then solve for x. It's a pretty straightforward process, but let's break it down step by step to make sure we nail it.

Step 1: Set the Function Equal to Zero

The first thing we need to do is replace f(x) with 0. This gives us the equation:

0 = -2x + 5

This equation is the starting point for finding the zero of the function. We're essentially asking, "What value of x will make this equation true?"

Step 2: Isolate the Term with x

Next, we want to get the term with x (-2x in this case) by itself on one side of the equation. To do this, we can subtract 5 from both sides of the equation:

0 - 5 = -2x + 5 - 5

This simplifies to:

-5 = -2x

Now we have the x term isolated on the right side, which is a crucial step in solving for x.

Step 3: Solve for x

Finally, to get x by itself, we need to divide both sides of the equation by the coefficient of x, which is -2:

-5 / -2 = -2x / -2

This simplifies to:

2.5 = x

So, we've found that the value of x that makes f(x) = 0 is 2.5. This means that the function f(x) = -2x + 5 crosses the x-axis at the point (2.5, 0).

Verifying the Solution

It's always a good idea to double-check your work, especially in math. To verify that our solution, x = 2.5, is correct, we can plug it back into the original function and see if we get zero. Let's do it:

f(2.5) = -2(2.5) + 5

f(2.5) = -5 + 5

f(2.5) = 0

Ta-da! It works! When we plug in x = 2.5, the function evaluates to zero, which confirms that our solution is correct. This step is crucial because it ensures that you haven't made any mistakes in your calculations. By verifying your solution, you can be confident in your answer and avoid errors in subsequent steps or applications.

Graphical Interpretation

To really solidify our understanding, let's take a look at what this means graphically. Remember, the zero of a function is the point where the graph of the function intersects the x-axis. In our case, the function f(x) = -2x + 5 is a linear function, which means its graph is a straight line. The zero of the function, x = 2.5, corresponds to the x-coordinate of the point where this line crosses the x-axis.

If you were to plot the graph of f(x) = -2x + 5, you would see a line that slopes downwards from left to right. This is because the coefficient of x is negative (-2). The line would intersect the y-axis at the point (0, 5), which is the y-intercept of the function. And, most importantly, the line would intersect the x-axis at the point (2.5, 0), which is the zero of the function that we just calculated.

Visualizing the function's graph can provide a deeper understanding of what the zero represents. It's not just a numerical value; it's a specific point on the coordinate plane where the function's value is zero. This graphical interpretation can be particularly helpful when dealing with more complex functions, where the zeros might not be as easily found algebraically.

Applying the Concept to Other Functions

The process we used to find the zero of f(x) = -2x + 5 can be applied to other functions as well. The basic idea remains the same: set the function equal to zero and solve for x. However, the specific steps involved in solving for x might vary depending on the type of function. Let's take a quick look at how this concept applies to different types of functions:

  • Linear Functions: Like our example, linear functions are straightforward to solve. You typically just need to isolate the x term and then divide by its coefficient.
  • Quadratic Functions: Quadratic functions (functions of the form ax² + bx + c) can be solved using various methods, such as factoring, completing the square, or the quadratic formula. Each method has its advantages and disadvantages, and the best approach depends on the specific function.
  • Polynomial Functions: For higher-degree polynomial functions, finding the zeros can be more challenging. You might need to use techniques like synthetic division or numerical methods to approximate the zeros.
  • Trigonometric Functions: Trigonometric functions like sine, cosine, and tangent have zeros at specific intervals. Finding these zeros often involves using trigonometric identities and understanding the periodic nature of the functions.

The key takeaway here is that while the method for solving for x might change, the fundamental concept of setting the function equal to zero remains the same. So, whether you're dealing with a simple linear function or a more complex trigonometric one, you can always start by setting the function equal to zero and then applying the appropriate techniques to solve for x.

Conclusion

Alright, guys, we've covered a lot in this guide! We've explored what the zero of a function means, why it's important, and how to find it. We walked through a step-by-step solution for the function f(x) = -2x + 5, verified our answer, and even looked at the graphical interpretation. We also discussed how this concept applies to other types of functions. Finding the zero of a function is a fundamental skill in mathematics, and it's one that you'll use again and again in your studies and in real-world applications. So, keep practicing, and you'll become a pro at finding those zeros in no time!

Remember, the zero of a function is just the beginning. Once you master this concept, you'll be able to tackle more advanced topics like finding the roots of equations, analyzing the behavior of functions, and solving optimization problems. So, keep exploring, keep learning, and most importantly, keep having fun with math!