Calculating Function Values: H(x) = 8^x

Hey everyone! Today, we're diving into the world of functions, specifically focusing on the function h(x) = 8^x. Our mission, should we choose to accept it, is to calculate the value of h(x) for various x-values. It's like a fun little puzzle where we plug in numbers and see what pops out. Think of h(x) as a machine: you put an x in, and the machine spits out a result based on the rule 8^x. This means we take the number 8 and raise it to the power of x. Let's get started and figure out how this function works. We will find out what happens when we use negative, zero and positive numbers for the $x$ values. Get ready to flex those math muscles! We will be using the table to input the numbers.

Understanding the Function h(x) = 8^x

Alright, before we jump into the calculations, let's make sure we're all on the same page about what the function h(x) = 8^x actually means. In this function, x is the input, and h(x) is the output. The rule 8^x tells us to take the number 8 (our base) and raise it to the power of whatever x is (our exponent). Remember, the exponent tells us how many times to multiply the base by itself. For example, if x is 2, then h(2) = 8^2 = 8 * 8 = 64. This is crucial to grasp because it is the foundation of our calculations. When the exponent is positive, the answer is pretty straightforward. When it gets negative or zero, it requires us to understand some special rules. Now, let's break down this function step by step. We'll start with negative values, then zero, and finally positive values for x. Remember, understanding the fundamentals is the key to mastering any concept. So, let us get a grip on this function as we try to solve all the values. We can also use a calculator, but it's always good to understand the principles behind it.

Calculating h(x) for Negative x-values

Now, let's roll up our sleeves and tackle negative x-values. When x is negative, we're dealing with negative exponents. For example, if we have x = -1, then h(-1) = 8^(-1). But wait, what does a negative exponent even mean? Well, a negative exponent tells us to take the reciprocal of the base raised to the positive version of the exponent. So, 8^(-1) is the same as 1 / 8^1, which simplifies to 1/8. Similarly, if x = -2, then h(-2) = 8^(-2), which is the same as 1 / 8^2, or 1 / 64. This is very important. Always remember that, a^(-n) = 1/a^n. When dealing with negative values, we are using the concept of reciprocals. So, calculating these values involves understanding how negative exponents change the equation. The bigger the negative number, the smaller the fraction. Let us keep these key rules in mind as we complete the table. We need to remember how exponents work, and also how to calculate them when they are negative. So, it's not as hard as it seems, just take your time, and follow the math.

Calculating h(x) for x = 0

Let's keep the momentum going by figuring out what happens when x = 0. This is a crucial case because it introduces us to a fundamental rule of exponents: Anything raised to the power of zero equals 1. No matter what the base is – 8, 10, 1000, or even a negative number – if it's raised to the power of 0, the result is always 1. Thus, if x = 0, then h(0) = 8^0 = 1. This is a simple but essential concept. Understanding this rule helps you deal with all kinds of exponential problems. This holds true regardless of the base. This rule simplifies many equations, making calculations quicker. So, next time you see something raised to the power of zero, you know the answer immediately. This is one of the essential rules to remember. You will see that this is a very interesting concept, and also is the simplest to solve.

Calculating h(x) for Positive x-values

Finally, let's address the scenario where x is positive. This is probably the easiest case. When x is a positive integer, it just means we multiply the base (8 in our case) by itself x times. For instance, if x = 1, then h(1) = 8^1 = 8. If x = 2, then h(2) = 8^2 = 8 * 8 = 64. As x increases, the value of h(x) grows exponentially. That's why we call it an exponential function. The rate of growth is rapid. Understanding this is key to understanding how exponential functions work. We are just using the concept of exponents here. We have to multiply the base to itself the number of times shown by the exponent. The concept is pretty simple and we can apply them to solve any type of values. This is also how we determine any exponential growth such as compound interest, or population growth. By seeing the output, you can see how fast the value grows.

Populating the Table

Alright, guys, now that we've covered all the scenarios – negative x, zero, and positive x – let's fill in that table and find the final answers. Remember all the steps we discussed and let's go over it again.

So there you have it! We've successfully calculated h(x) for each x-value in the table. By going through the table, we were able to compute the values for h(x). We applied our knowledge of exponents and had fun while doing it.

Conclusion

In conclusion, we have learned how to calculate the value of the exponential function h(x) = 8^x for different values of x. We covered the cases where the exponent is negative, zero and positive. We also saw how important the understanding of the rules for exponents. The main take away from this is how to approach the different types of exponents, and remember that any number to the power of zero is always 1. Great work everyone!