Compute Function Value F(x) = 3 * (1/2)^x

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Hey guys! Are you ready to dive into the exciting world of functions? Today, we're going to tackle a specific function and break down how to compute its value. We'll be focusing on the function f(x) = 3 * (1/2)^x. Don't worry if this looks a bit intimidating at first – we'll take it step by step and you'll be a pro in no time!

Understanding the Function f(x) = 3 * (1/2)^x

Before we jump into calculations, let's make sure we understand what this function actually means. At its heart, this is an exponential function. Exponential functions are characterized by a constant base raised to a variable exponent. In our case, the base is 1/2 and the exponent is x. The "3" in front is simply a constant multiplier that scales the entire function.

To really grasp this, let's break it down further:

  • f(x): This is the name of our function. It tells us that the output of the function depends on the input x. Think of it as a machine: you put x in, and f(x) comes out.
  • 3: This is the constant multiplier. It means that whatever value we get from the exponential part, we'll multiply it by 3.
  • (1/2)^x: This is the exponential part. It means we're taking the base 1/2 and raising it to the power of x. Remember that raising something to a power means multiplying it by itself that many times. For example, (1/2)^2 = (1/2) * (1/2) = 1/4.
  • x: This is the input variable. We can plug in any number for x, and the function will give us a corresponding output.

Key Concepts to Remember

  • Exponential Functions: Functions where a constant base is raised to a variable exponent. They often model growth or decay.
  • Base: The constant being raised to a power (in our case, 1/2).
  • Exponent: The variable power (in our case, x).
  • Constant Multiplier: A number that scales the entire function (in our case, 3).

Why is this important? Understanding exponential functions is crucial in many areas of mathematics, science, and finance. They pop up in everything from population growth and radioactive decay to compound interest and the spread of diseases. By mastering this basic function, you're building a strong foundation for tackling more complex problems down the road.

Now that we have a solid understanding of the function itself, let's move on to the main task at hand: computing its value for a specific input. In this case, we're asked to find f(-1). This means we need to plug in x = -1 into our function and see what comes out.

Calculating f(-1) Step-by-Step

Okay, let's get our hands dirty and calculate f(-1). This is where the magic happens! Here's how we'll do it, step by step:

  1. Substitute x with -1: This is the first and most crucial step. We're replacing the variable x in our function with the specific value we're interested in, which is -1. So, we rewrite our function as:

    f(-1) = 3 * (1/2)^(-1)

    Notice how we've simply replaced x with -1. Easy peasy!

  2. Deal with the negative exponent: Now, we encounter a negative exponent. Negative exponents can seem a bit tricky, but they're actually quite straightforward to handle. Remember this rule: a^(-n) = 1 / a^n. In other words, a negative exponent means we take the reciprocal of the base and raise it to the positive version of the exponent.

    Applying this rule to our function, we get:

    (1/2)^(-1) = 1 / (1/2)^1

    And since anything raised to the power of 1 is just itself, we have:

    1 / (1/2)^1 = 1 / (1/2)

    Dividing by a fraction is the same as multiplying by its reciprocal. So,

    1 / (1/2) = 1 * (2/1) = 2

    Therefore, (1/2)^(-1) = 2. We've successfully conquered the negative exponent!

  3. Substitute the simplified exponent back into the function: Now that we know (1/2)^(-1) = 2, we can plug this value back into our expression for f(-1):

    f(-1) = 3 * 2

  4. Perform the final multiplication: This is the easy part! We simply multiply 3 by 2:

    3 * 2 = 6

    And there you have it! We've calculated the value of the function at x = -1.

  5. State the result: f(-1) = 6

Recap of the Steps

  1. Substitute x with -1: f(-1) = 3 * (1/2)^(-1)
  2. Deal with the negative exponent: (1/2)^(-1) = 2
  3. Substitute the simplified exponent back into the function: f(-1) = 3 * 2
  4. Perform the final multiplication: 3 * 2 = 6
  5. State the result: f(-1) = 6

See? It's not so scary when you break it down into manageable steps. Understanding the rules of exponents is key here. The negative exponent rule is a fundamental concept in algebra, so make sure you have a good grasp of it. Now, let's summarize our findings and discuss the significance of our result.

Summarizing the Result and Its Significance

We've successfully computed the value of the function f(x) = 3 * (1/2)^x at x = -1. Our calculation showed that f(-1) = 6. This means that when we input -1 into our function