Solving F(x) = 5x + 10: Find The Zero

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into a super common but sometimes tricky topic in mathematics: finding the zero of a function. Specifically, we're tackling the linear function $f(x) = 5x + 10$. Finding the zero of a function is all about figuring out what input value, or $x$-value, makes the function's output, $f(x)$, equal to zero. Think of it as finding where the function crosses the $x$-axis on a graph. It’s a fundamental concept that pops up all over the place in math, science, and engineering, so understanding it really well is a game-changer. We'll break down exactly what a "zero" is, why it's important, and then walk through solving for the zero of $f(x) = 5x + 10$. We've got some multiple-choice options to help us nail it down: A. 2, B. -10, C. 10, D. -2. Let's get this bread and figure out which one is the correct answer!

What Exactly is a "Zero" of a Function?

So, what are we even looking for when we talk about the "zero" of a function, guys? In simple terms, the zero of a function, also known as a root or a x-intercept, is the value of the input variable (usually $x$) that makes the function's output ($f(x)$ or $y$) equal to zero. Mathematically, if we have a function $f(x)$, we are trying to find the value(s) of $x$ such that $f(x) = 0$. This concept is super important because it tells us where the graph of the function intersects the $x$-axis. For a linear function like $f(x) = 5x + 10$, there will be exactly one zero, unless the line is perfectly horizontal and not on the $x$-axis (in which case there are no zeros) or it is the $x$-axis itself (in which case every point is a zero). Understanding this helps us solve equations, model real-world phenomena, and much more. For instance, in physics, finding the zero of a function might represent the time when an object hits the ground or when a population reaches zero. In economics, it could be the break-even point where costs equal revenue. So, even though it sounds basic, this idea of finding where $f(x) = 0$ is incredibly powerful. It's the foundation for solving quadratic equations, analyzing polynomial behavior, and understanding the stability of systems. We're going to use this core definition to solve our specific problem, $f(x) = 5x + 10$, and find that specific $x$-value that makes the whole expression equal to zero. Stick around, and we'll break it down step-by-step so it makes perfect sense.

Solving for the Zero of $f(x) = 5x + 10$

Alright, let’s get down to business and actually solve for the zero of our function, $f(x) = 5x + 10$. Remember, finding the zero means we want to find the value of $x$ that makes $f(x) = 0$. So, the very first step is to set the function equal to zero:

$$5x + 10 = 0$$

Now, our goal is to isolate $x$ on one side of the equation. This is a standard linear equation, and we can solve it using basic algebraic operations. We want to get $x$ by itself. First, we need to get the term with $x$ ($5x$) by itself on one side. To do this, we need to move the constant term ($+10$) to the other side of the equation. We can achieve this by subtracting 10 from both sides of the equation:

$$5x + 10 - 10 = 0 - 10$$

This simplifies to:

$$5x = -10$$

Awesome, we're one step closer! Now, $x$ is being multiplied by 5. To get $x$ completely alone, we need to undo the multiplication. The opposite of multiplying by 5 is dividing by 5. So, we divide both sides of the equation by 5:

$$\frac{5x}{5} = \frac{-10}{5}$$

This gives us our final answer:

$$x = -2$$

So, the zero of the function $f(x) = 5x + 10$ is $x = -2$. This means that when $x$ is $-2$, the function $f(x)$ evaluates to 0. Let's quickly check this: $f(-2) = 5(-2) + 10 = -10 + 10 = 0$. It works perfectly!

Evaluating the Options

Now that we've worked through the problem and found our answer, let's look at the multiple-choice options provided and see which one matches our result. We found that the zero of $f(x) = 5x + 10$ is $x = -2$. Let's check the options:

• A. 2: If we plug in $x = 2$, we get $f(2) = 5(2) + 10 = 10 + 10 = 20$. This is not 0, so option A is incorrect.
• B. -10: If we plug in $x = -10$, we get $f(-10) = 5(-10) + 10 = -50 + 10 = -40$. This is not 0, so option B is incorrect.
• C. 10: If we plug in $x = 10$, we get $f(10) = 5(10) + 10 = 50 + 10 = 60$. This is not 0, so option C is incorrect.
• D. -2: If we plug in $x = -2$, we get $f(-2) = 5(-2) + 10 = -10 + 10 = 0$. This is exactly what we're looking for! So, option D is the correct answer.

Conclusion

There you have it, guys! We successfully found the zero of the function $f(x) = 5x + 10$. By setting the function equal to zero and solving for $x$, we determined that $x = -2$ is the value that makes the function output zero. This corresponds to option D in our multiple-choice list. Remember, finding the zero of a function is a fundamental skill in mathematics, essential for understanding graphs, solving equations, and modeling real-world problems. Whether you're dealing with linear functions like this one or more complex polynomials and beyond, the core idea remains the same: find the input that yields an output of zero. Keep practicing these types of problems, and you'll be a math whiz in no time! Thanks for tuning into Plastik Magazine. Stay curious and keep exploring the amazing world of mathematics!