To find the x-intercept(s) of a function, we need to determine the points where the graph of the function crosses the x-axis. At these points, the y-coordinate is always zero. Therefore, to find the x-intercept(s), we need to find the real solutions to the equation h(x) = 0. This article will guide you through the process using the example function h(x) = 3x + 7. Let's dive in and make this crystal clear, guys!
Understanding X-Intercepts
Before we jump into the solution, let's make sure we're all on the same page about what an x-intercept actually is. The x-intercept is the point where a graph intersects the x-axis. Think of it as the spot where the line or curve "lands" on the horizontal axis. At this point, the y-value is always zero. This is super important to remember because it’s the key to finding x-intercepts algebraically.
Why is the y-value zero? Well, the x-axis itself is defined as the line where y = 0. So, any point on the x-axis has a y-coordinate of 0. When we're looking for x-intercepts, we're essentially looking for the x-values that make the function equal to zero. These x-values are also known as the roots or zeros of the function. So, when someone asks you to find the roots or zeros, they're essentially asking you to find the x-intercepts!
In the context of a linear equation like h(x) = 3x + 7, the x-intercept represents the point where the line crosses the x-axis. For more complex functions, there might be multiple x-intercepts, meaning the graph crosses the x-axis at several points. But for a simple linear equation, we'll typically find just one x-intercept. Finding this point is crucial in many mathematical and real-world applications. For example, in economics, the x-intercept of a supply or demand curve can represent the price at which there is no supply or demand. In physics, it could represent the point where an object's trajectory intersects the ground. So, understanding how to find x-intercepts is a fundamental skill in many fields.
The Equation: h(x) = 3x + 7
Now, let's focus on the specific equation we're working with: h(x) = 3x + 7. This is a linear equation, which means its graph is a straight line. Linear equations are among the simplest types of equations to work with, and finding their x-intercepts is a straightforward process. The equation is in slope-intercept form, although we don't explicitly need to know that to solve it. The key thing is that we have a function, h(x), that we want to set equal to zero to find our x-intercept.
In this equation, '3' is the coefficient of 'x', which represents the slope of the line. The '+ 7' is the constant term, which represents the y-intercept (the point where the line crosses the y-axis). However, for our task of finding the x-intercept, we're primarily concerned with setting the entire function h(x) equal to zero. Remember, we're looking for the x-value that makes the function equal to zero, because that's where the graph crosses the x-axis.
Understanding the components of this equation helps us visualize what we're doing. We're essentially asking: "What value of 'x' do I need to plug into this equation so that the result is zero?" This is a fundamental question in algebra, and the process of answering it is what we'll go through step-by-step. The good news is that for linear equations like this, there's usually only one solution, meaning there's only one x-intercept. This makes our task a bit simpler and more direct. So, let's get to the heart of the problem: substituting h(x) = 0 and solving for 'x'.
Step 1: Substitute h(x) = 0
The first step in finding the x-intercept is to substitute h(x) with 0 in the equation. This is because, as we discussed earlier, the x-intercept occurs where the y-value (which is represented by h(x) in this case) is zero. So, we're essentially saying, "Let's find the x-value when the function's output is zero." This is a crucial step because it sets up the equation we need to solve for 'x'.
So, we take our original equation, h(x) = 3x + 7, and replace h(x) with 0. This gives us the equation 0 = 3x + 7. Now we have a simple algebraic equation with one variable, 'x', which we need to solve. This substitution is the bridge between the concept of the x-intercept and the algebraic process of finding it. It transforms the problem from a geometrical one (finding where the graph crosses the x-axis) to an algebraic one (solving an equation). This is a common technique in mathematics – translating problems into different forms to make them easier to solve.
By making this substitution, we've created a clear path forward. We now have an equation that we can manipulate using algebraic techniques to isolate 'x' and find its value. This value will be the x-coordinate of our x-intercept. The next steps will involve using basic algebraic operations to get 'x' by itself on one side of the equation. This is a fundamental skill in algebra, and it's essential for solving a wide range of problems. So, let's move on to the next step and see how we can isolate 'x'.
Step 2: Solve for x
Now that we have the equation 0 = 3x + 7, our goal is to isolate 'x' on one side of the equation. This means we want to get 'x' by itself, so we know its value. To do this, we'll use basic algebraic operations, specifically inverse operations, to undo the operations that are being performed on 'x'.
The first thing we need to do is get rid of the '+ 7' on the right side of the equation. To do this, we'll subtract 7 from both sides of the equation. Remember, whatever we do to one side of the equation, we must do to the other side to maintain the equality. This gives us: 0 - 7 = 3x + 7 - 7, which simplifies to -7 = 3x. Great! We've made progress in isolating 'x'.
Next, we need to get rid of the '3' that's multiplying 'x'. To do this, we'll divide both sides of the equation by 3. This gives us: -7 / 3 = 3x / 3, which simplifies to x = -7/3. And there we have it! We've solved for 'x'. The value of x that makes the equation true is -7/3. This is the x-coordinate of our x-intercept. So, we've successfully found the value of x where the function h(x) equals zero.
This process of isolating a variable is a fundamental skill in algebra, and it's used in countless applications. By using inverse operations, we can systematically undo the operations that are being performed on the variable until we have it by itself. This allows us to determine the value of the variable that satisfies the equation. In this case, we've found the x-value that makes our function equal to zero, which is exactly what we needed to find the x-intercept.
Step 3: Express the X-Intercept as a Coordinate Point
We've found that x = -7/3 is the solution to the equation 0 = 3x + 7. This means that when x is -7/3, the function h(x) equals zero. But to fully express the x-intercept, we need to write it as a coordinate point. Remember, a coordinate point has two values: an x-value and a y-value. We already know the x-value is -7/3. What about the y-value?
Well, remember that the x-intercept is the point where the graph crosses the x-axis. At any point on the x-axis, the y-value is always zero. This is a crucial concept that we discussed earlier. So, the y-coordinate of our x-intercept is 0. Therefore, we can express the x-intercept as the coordinate point (-7/3, 0). This is the point where the line represented by the equation h(x) = 3x + 7 crosses the x-axis.
Writing the x-intercept as a coordinate point is important because it clearly shows both the x and y values at the point of intersection. It provides a complete picture of where the graph crosses the x-axis. In this case, the point (-7/3, 0) tells us that the line crosses the x-axis at x = -7/3 and y = 0. This is a standard way of representing intercepts and points in the coordinate plane.
Conclusion
So, to recap, we've successfully found the x-intercept of the function h(x) = 3x + 7 by following a few simple steps. First, we substituted h(x) with 0, giving us the equation 0 = 3x + 7. Then, we solved for 'x' by using inverse operations, which led us to x = -7/3. Finally, we expressed the x-intercept as a coordinate point, which is (-7/3, 0). This process demonstrates a fundamental concept in algebra and is a crucial skill for anyone studying mathematics.
Finding x-intercepts is not just a mathematical exercise; it has practical applications in various fields. Whether you're analyzing graphs, solving equations, or modeling real-world scenarios, understanding how to find x-intercepts is essential. The steps we've outlined here can be applied to any function, although the complexity of the equation might vary. For linear equations like the one we worked with, the process is relatively straightforward. However, for more complex functions, you might need to use different techniques, such as factoring, the quadratic formula, or numerical methods.
The key takeaway here is the understanding of what an x-intercept represents and how to find it algebraically. Remember, the x-intercept is the point where the graph crosses the x-axis, and the y-value at this point is always zero. By setting the function equal to zero and solving for 'x', we can find the x-coordinate of the x-intercept. And by expressing the result as a coordinate point, we provide a complete and clear representation of the x-intercept. So keep practicing, and you'll become a pro at finding x-intercepts in no time!
