Completing Tables: F(x) = 3x + 1 & G(x) = 5^(x-2)

Hey guys! Today, we're going to dive into completing tables of values for two different functions: a linear function, f(x) = 3x + 1, and an exponential function, g(x) = 5^(x-2). Understanding how to do this is super important in mathematics because it helps us visualize and analyze how functions behave. We'll break it down step-by-step, making it easy to follow along. Let's get started!

Understanding Function Tables

Before we jump into the calculations, let's quickly recap what a function table actually represents. A function table is essentially a way to organize the input (x) and output (f(x) or g(x)) values of a function. You plug in a specific value for x, do the math according to the function's rule, and then you get a corresponding output value. These tables are incredibly useful for plotting graphs, identifying patterns, and understanding the relationship between variables. They give us a clear, visual snapshot of how a function transforms different inputs into outputs.

When dealing with function tables, it's important to remember that each x value corresponds to a unique y value (which is the same as f(x) or g(x) in this context). This is the fundamental concept of a function – for every input, there's only one output. Filling out a table helps us see this relationship in action. We'll be using substitution, a core algebraic technique, to find these output values. Substitution simply means replacing a variable (x in our case) with a specific number and then simplifying the expression. It’s a basic but powerful tool that’s used throughout mathematics and science. So, let's warm up those substitution skills and get ready to fill in our table!

Completing the Table for f(x) = 3x + 1

Let's start with the linear function f(x) = 3x + 1. Remember, a linear function will always create a straight line when graphed, and understanding this type of function is essential. We need to find the values of f(x) for x = 2, 3, 4, and 5. This means we'll substitute each of these x values into the function and calculate the result. It's like a mathematical recipe – we have a set of instructions (the function) and we're just plugging in different ingredients (x values) to see what dish we get (the f(x) value).

• For x = 2:
  • f(2) = 3(2) + 1
  • f(2) = 6 + 1
  • f(2) = 7
• For x = 3:
  • f(3) = 3(3) + 1
  • f(3) = 9 + 1
  • f(3) = 10
• For x = 4:
  • f(4) = 3(4) + 1
  • f(4) = 12 + 1
  • f(4) = 13
• For x = 5:
  • f(5) = 3(5) + 1
  • f(5) = 15 + 1
  • f(5) = 16

Now, we have all the f(x) values for the given x values. Notice how the f(x) values increase by 3 each time x increases by 1? That's a key characteristic of linear functions – they have a constant rate of change, which is the slope of the line. Filling out these values helps us to see the consistent pattern of the linear function, and it reinforces our understanding of how these functions work. So, with the f(x) values calculated, we’re halfway to completing our table!

Completing the Table for g(x) = 5^(x-2)

Next up, we have the exponential function g(x) = 5^(x-2). Exponential functions behave very differently from linear functions. Instead of a constant rate of change, they grow (or decay) at an increasing rate. This means their graphs curve upwards or downwards dramatically. Calculating values for an exponential function can sometimes seem a bit trickier, but the core concept is still the same: substitute the x value into the function and simplify. The exponent here is x-2, so we need to evaluate this first before raising 5 to that power. This requires us to follow the order of operations carefully, ensuring we perform the subtraction in the exponent before the exponentiation itself. Let's find the values of g(x) for x = 2, 3, 4, and 5.

• For x = 2:
  • g(2) = 5^(2-2)
  • g(2) = 5^0
  • g(2) = 1 (Remember, any non-zero number raised to the power of 0 is 1)
• For x = 3:
  • g(3) = 5^(3-2)
  • g(3) = 5^1
  • g(3) = 5
• For x = 4:
  • g(4) = 5^(4-2)
  • g(4) = 5^2
  • g(4) = 25
• For x = 5:
  • g(5) = 5^(5-2)
  • g(5) = 5^3
  • g(5) = 125

Notice how the g(x) values are increasing much faster than the f(x) values? That’s the power of exponential growth! Each time x increases by 1, g(x) is multiplied by 5. This rapid growth is a key feature of exponential functions, and it’s why they’re used to model things like population growth and compound interest. By calculating these values, we're building a strong understanding of how exponential functions work and how they differ from linear functions.

The Completed Table

Alright, we've done all the hard work! Now, let's put it all together and show our completed table. This table neatly summarizes the f(x) and g(x) values we calculated for each x value. Seeing the table in its entirety allows us to easily compare the behavior of the two functions and visualize their patterns. A well-organized table is a powerful tool for understanding the relationship between the input and output of functions. Here's what it looks like:

By examining this table, you can clearly see the difference between the linear and exponential functions. f(x) increases steadily, while g(x) increases dramatically. This comparison highlights the fundamental nature of these two types of functions and their diverse applications in the real world. You can now confidently use these values to plot graphs of the functions or use them in further analysis.

Conclusion

So there you have it! We've successfully completed the table of values for both the linear function f(x) = 3x + 1 and the exponential function g(x) = 5^(x-2). We walked through the process of substituting x values into each function, calculating the corresponding f(x) and g(x) values, and then organizing those values into a clear and concise table. Remember, practice makes perfect, so keep working on these types of problems to solidify your understanding.

Completing tables like this is a fundamental skill in mathematics. It not only reinforces your understanding of functions but also provides a solid foundation for more advanced topics like graphing, calculus, and mathematical modeling. Being able to confidently calculate and interpret these values will serve you well in your mathematical journey. Plus, you guys now have a deeper insight into how different types of functions behave, which is super cool! Keep up the awesome work!