Function Values: Find F(1) And F(6) For F(x) = X + 5
Alright, guys, let's dive into a super straightforward math problem that's all about understanding functions. We're given the function f(x) = x + 5, and our mission is to find the values of f(1) and f(6). In essence, this means we need to substitute '1' and '6' for 'x' in the function and see what we get. It’s like a simple machine where you put in a number, and it spits out another number based on the rule defined by the function. So, grab your thinking caps, and let’s get started!
Understanding the Function
Before we jump into calculations, let's make sure we fully grasp what the function f(x) = x + 5 is telling us. A function is essentially a rule that assigns a unique output to each input. In this case, our input is 'x', and the rule is to add 5 to whatever 'x' is. So, if x is 2, then f(2) would be 2 + 5, which equals 7. Simple, right? The beauty of functions is their predictability; for every 'x' you plug in, you'll always get the same corresponding output. This makes them incredibly useful in mathematics, computer science, and many other fields.
Functions are the fundamental blocks for modeling relationships between variables. They provide a structured way to describe how one quantity changes in relation to another. Whether we're talking about the trajectory of a ball thrown in the air, the growth of a bacteria population, or the cost of producing a certain number of items, functions allow us to represent these real-world phenomena mathematically. Understanding how to evaluate functions for different inputs is a core skill in mathematics and is crucial for more advanced topics like calculus and differential equations. Moreover, functions are not limited to simple algebraic expressions like f(x) = x + 5. They can involve trigonometric functions, exponential functions, logarithmic functions, and even piecewise definitions where different rules apply for different ranges of 'x'. As you delve deeper into mathematics, you'll encounter a vast array of functions, each with its unique properties and applications. So, mastering the basics now will set you up for success in the future.
Now that we have a good understanding of the function, let’s use this knowledge to find the values of f(1) and f(6).
Calculating f(1)
Okay, let's find f(1). This means we're going to replace 'x' with '1' in our function. So, instead of f(x) = x + 5, we'll have f(1) = 1 + 5. And what does that equal? That's right, it's 6! So, f(1) = 6. See how easy that was? It's just a matter of plugging in the value and doing the math. No sweat!
To further illustrate, let's break it down step-by-step. We start with the function f(x) = x + 5. Our goal is to find the value of the function when x is equal to 1. This is denoted as f(1). To do this, we simply substitute 1 for every instance of x in the function's expression. So, we replace x with 1 in the equation f(x) = x + 5, which gives us f(1) = 1 + 5. Now, we just need to perform the addition. 1 plus 5 equals 6. Therefore, f(1) = 6. This means that when the input to the function is 1, the output is 6. Understanding this process is crucial because it's the foundation for evaluating functions with more complex expressions and different types of inputs.
Now, let's consider a slightly different scenario. What if the function was f(x) = 2x + 3? In this case, to find f(1), we would substitute 1 for x in the expression 2x + 3. This gives us f(1) = 2(1) + 3. Following the order of operations, we would first perform the multiplication: 2 times 1 equals 2. Then, we would add 3 to the result: 2 plus 3 equals 5. Therefore, f(1) = 5 for the function f(x) = 2x + 3. As you can see, the process is the same regardless of the complexity of the function's expression. The key is to carefully substitute the input value for every instance of x and then follow the order of operations to simplify the expression and find the output value. With practice, you'll become proficient at evaluating functions for any input value.
Calculating f(6)
Alright, moving on to f(6). This time, we replace 'x' with '6' in our function f(x) = x + 5. So, we get f(6) = 6 + 5. What's 6 + 5? It's 11! Therefore, f(6) = 11. We're on a roll, guys! Finding the values of a function is all about substitution and simple arithmetic.
Let's take another look at the function f(x) = x + 5. This function represents a straight line when graphed on a coordinate plane. The x variable represents the input or the horizontal coordinate, and the f(x) variable represents the output or the vertical coordinate. The equation f(x) = x + 5 tells us that for every increase of 1 in the x value, the f(x) value increases by 1 as well. The constant term, 5, represents the y-intercept of the line, which is the point where the line crosses the vertical axis. In this case, the line crosses the y-axis at the point (0, 5). Understanding the graphical representation of a function can provide valuable insights into its behavior and properties.
We found that f(1) = 6 and f(6) = 11. These two points, (1, 6) and (6, 11), lie on the line represented by the function f(x) = x + 5. If we were to plot these points on a graph and draw a line through them, we would see that the line has a slope of 1 and a y-intercept of 5, which confirms our understanding of the function's equation. Furthermore, we can use these two points to calculate the slope of the line using the formula: slope = (y2 - y1) / (x2 - x1). Plugging in the coordinates of the two points, we get: slope = (11 - 6) / (6 - 1) = 5 / 5 = 1. This confirms that the slope of the line is indeed 1, as indicated by the coefficient of the x term in the function's equation.
Final Answer
So, to wrap it up, we found that f(1) = 6 and f(6) = 11 for the function f(x) = x + 5. That's all there is to it! Remember, when you're asked to find the value of a function at a specific point, just replace the variable with that point and simplify. You got this!
In summary, evaluating functions is a fundamental skill in mathematics. By understanding the concept of a function as a rule that assigns a unique output to each input, you can easily find the value of a function for any given input. The process involves substituting the input value for the variable in the function's expression and then simplifying the expression according to the order of operations. Whether the function is a simple algebraic expression or a more complex one involving trigonometric, exponential, or logarithmic functions, the basic principle remains the same. With practice, you'll become proficient at evaluating functions and using them to model and solve real-world problems. So, keep practicing, and you'll be amazed at how far you can go!