Solving F(x) = G(x) When F(x) Is -3x + 4 And G(x) Is 2

Hey guys! Today, we're diving into a fun little math problem where we need to find the value of x that makes two functions, f(x) and g(x), equal to each other. We've got f(x) = -3x + 4 and g(x) = 2. Our mission, should we choose to accept it (and we do!), is to figure out what x needs to be so that f(x) gives us the same result as g(x). Let's jump right in and break this down step by step.

Understanding the Functions

Before we start solving, it’s super important to understand what these functions are telling us. Think of a function like a machine: you put a number in (x), and the machine does some stuff to it and spits out another number (f(x) or g(x)).

Delving into f(x) = -3x + 4

In the case of f(x) = -3x + 4, it’s a linear function. That means if we were to graph it, it would look like a straight line. This function tells us to take any number x, multiply it by -3, and then add 4. So, if we put in x = 1, we get f(1) = -3(1) + 4 = 1. If we put in x = 2, we get f(2) = -3(2) + 4 = -2. See how it works? Understanding this is the first key to unlocking the problem.

Linear functions like this are everywhere in math and real life. They help us model relationships where things change at a constant rate. For example, imagine you're saving money. If you start with $10 and save $5 every week, that's a linear relationship. The function could be something like savings(weeks) = 5 * weeks + 10. Knowing the slope (-3 in our case) tells us how steeply the line goes up or down. A negative slope means the line goes downwards as x increases, which is important for visualizing what's happening with f(x). The y-intercept (4 in our case) tells us where the line crosses the vertical axis, which is the value of the function when x is zero.

Understanding g(x) = 2

Now, let’s talk about g(x) = 2. This might seem simpler, and that’s because it is! This is a constant function. No matter what number we put in for x, g(x) will always be 2. Always! If we graphed this, it would be a horizontal line at y = 2. So, g(0) = 2, g(100) = 2, g(-5) = 2 – it just doesn’t change. This function represents a fixed value, which can be useful in many situations where something stays the same regardless of the input.

Think about it like this: imagine you have a job that pays a flat rate of $2 per hour, no matter how many hours you work. The function representing your pay would be constant, just like g(x). Constant functions are essential in various mathematical and real-world scenarios, especially as baseline values or fixed quantities.

Setting f(x) Equal to g(x)

Okay, now for the main event! We need to find the x that makes f(x) equal to g(x). Mathematically, we write this as:

f(x) = g(x)

This means we’re looking for the point where the “machine” f(x) spits out the same number as the “machine” g(x). In graphical terms, we’re looking for the point where the line representing f(x) intersects the horizontal line representing g(x). This point of intersection gives us the x value that satisfies our equation.

So, we take our functions and set them equal to each other:

-3x + 4 = 2

This equation is the heart of our problem. It’s saying, “For what x value will -3x + 4 be the same as 2?” Now, it’s just a matter of solving for x.

Solving the Equation

To solve for x, we need to isolate it on one side of the equation. This involves a bit of algebraic maneuvering, but don't worry, it's totally doable! Here’s how we can do it:

Step 1: Subtract 4 from Both Sides

Our goal is to get the term with x by itself. To do this, we can subtract 4 from both sides of the equation. Remember, what we do to one side, we must do to the other to keep the equation balanced. It’s like a scale – if you take something off one side, you need to take the same amount off the other to keep it level.

-3x + 4 - 4 = 2 - 4

This simplifies to:

-3x = -2

Now, we’re one step closer! We’ve got -3x on one side, which is much simpler than -3x + 4.

Step 2: Divide Both Sides by -3

Now we need to get x all by itself. It’s currently being multiplied by -3, so to undo that, we need to divide both sides of the equation by -3. Again, keeping the balance is key!

-3x / -3 = -2 / -3

This simplifies to:

x = 2/3

And there we have it! We’ve solved for x. The value of x that makes f(x) = g(x) is 2/3.

Verifying the Solution

But wait, we're not quite done yet! It’s always a good idea to double-check our work to make sure we didn’t make any mistakes. This is especially important in math because a small error early on can throw off the whole answer. To verify our solution, we can plug x = 2/3 back into both f(x) and g(x) and see if we get the same result.

Checking with f(x)

Let’s start with f(x) = -3x + 4:

f(2/3) = -3(2/3) + 4

First, we multiply -3 by 2/3:

-3 * (2/3) = -2

Then we add 4:

-2 + 4 = 2

So, f(2/3) = 2.

Checking with g(x)

Now let’s check g(x) = 2. Remember, g(x) is a constant function, so no matter what we put in for x, it always gives us 2.

g(2/3) = 2

The Verdict

We found that f(2/3) = 2 and g(2/3) = 2. They’re the same! This confirms that our solution, x = 2/3, is correct. We’ve successfully found the value of x that makes the two functions equal.

Graphical Interpretation

Now, let's bring in a visual perspective to make sure we understand this on a deeper level. Imagine we're plotting these functions on a graph. The function f(x) = -3x + 4 is a straight line with a negative slope, meaning it goes downwards as we move from left to right. It crosses the y-axis at the point (0, 4). On the other hand, g(x) = 2 is a horizontal line that runs straight across at the height of 2 on the y-axis.

The solution we found, x = 2/3, is actually the x-coordinate of the point where these two lines intersect. At this point, the y-values of both functions are the same, which is 2. So, if you were to draw these lines on a graph, you would see them cross each other precisely at the point where x is 2/3 and y is 2. This graphical representation gives us another way to confirm that our solution is correct and helps us visualize what it means for two functions to be equal.

Understanding the graphical side of functions can be super helpful because it gives you a visual check on your algebraic solutions. It’s like having a map to make sure you’re on the right path! Next time you're solving a problem like this, try sketching a quick graph – it might just make everything click.

Conclusion

So, to wrap it all up, we started with two functions, f(x) = -3x + 4 and g(x) = 2, and we wanted to find the value of x that makes them equal. We set f(x) equal to g(x), solved the resulting equation, and found that x = 2/3. We then verified our solution by plugging it back into both functions and confirmed that it works. We even took a peek at how this looks on a graph, reinforcing our understanding.

This type of problem is a fundamental concept in algebra, and mastering it opens the door to more complex topics. Keep practicing, and you'll become a function-solving pro in no time! Remember, math is like building with blocks – each concept builds upon the last, so getting the basics down solid is key. Great job, guys, and keep up the awesome work!