Inverse Of Log Function: Complete The Table

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Hey guys! Today, we're diving into the fascinating world of inverse functions, specifically focusing on the inverse of a logarithmic function. We've got a fun problem where we need to figure out some values for a table. The function we’re working with is $f(x) = \log_{0.5} x$, and its inverse is given by $f^{-1}(x) = 0.5^x$. Our mission, should we choose to accept it, is to find the values that complete the table for this inverse function. So, let's roll up our sleeves and get started!

Understanding Inverse Functions

Before we jump into the calculations, let's take a quick moment to make sure we're all on the same page about what inverse functions are. An inverse function is essentially a function that "undoes" what the original function does. Think of it like this: if you have a function that turns apples into apple juice, the inverse function would (hypothetically) turn apple juice back into apples. Pretty neat, right?

In mathematical terms, if we have a function $f(x)$, its inverse, denoted as $f^{-1}(x)$, has the property that $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$. This means that if you plug the output of a function into its inverse, you get back the original input. For our specific problem, we're given the logarithmic function $f(x) = \log_{0.5} x$ and its inverse exponential function $f^{-1}(x) = 0.5^x$. This inverse function is what we'll be using to complete our table.

Understanding inverse functions is crucial in many areas of mathematics and real-world applications. They help us solve equations, analyze relationships between variables, and even in fields like cryptography and computer science. For example, in cryptography, inverse functions are used to decrypt messages, turning encoded information back into its original form. This "undoing" property is what makes inverse functions so powerful and versatile.

In the context of our problem, we'll be using the inverse function $f^{-1}(x) = 0.5^x$ to find the corresponding y-values for given x-values. This process involves plugging in the x-values into the inverse function and calculating the result. By doing this for each x-value in our table, we'll be able to complete it and gain a better understanding of how the inverse function behaves. So, with this basic understanding under our belts, let's dive into the calculations!

Calculating Values for the Inverse Function Table

Okay, let's get down to business and calculate the missing values for our table. We're given the inverse function $f^{-1}(x) = 0.5^x$, and we need to find the corresponding y-values for the following x-values: -2, -1, 0, 1, and 2. We'll calculate each one step-by-step.

When x = -2

To find the value of the inverse function when x = -2, we substitute -2 into the function:

fβˆ’1(βˆ’2)=0.5βˆ’2f^{-1}(-2) = 0.5^{-2}

Remember that a negative exponent means we take the reciprocal of the base raised to the positive exponent. So,

0.5βˆ’2=(10.5)2=22=40. 5^{-2} = \left(\frac{1}{0.5}\right)^2 = 2^2 = 4

So, when x = -2, $f^{-1}(x) = 4$.

When x = -1

Next up, let's find the value when x = -1:

fβˆ’1(βˆ’1)=0.5βˆ’1f^{-1}(-1) = 0.5^{-1}

Again, we use the property of negative exponents:

0.5βˆ’1=10.5=20. 5^{-1} = \frac{1}{0.5} = 2

Thus, when x = -1, $f^{-1}(x) = 2$.

When x = 0

Now, let's tackle the case when x = 0:

fβˆ’1(0)=0.50f^{-1}(0) = 0.5^0

Any non-zero number raised to the power of 0 is 1. So,

0.50=10. 5^0 = 1

Therefore, when x = 0, $f^{-1}(x) = 1$.

When x = 1

Moving on, let's calculate the value when x = 1:

fβˆ’1(1)=0.51f^{-1}(1) = 0.5^1

Any number raised to the power of 1 is just the number itself. So,

0.51=0.50. 5^1 = 0.5

Hence, when x = 1, $f^{-1}(x) = 0.5$.

When x = 2

Finally, let's find the value when x = 2:

fβˆ’1(2)=0.52f^{-1}(2) = 0.5^2

This means we need to square 0.5:

0.52=0.5Γ—0.5=0.250. 5^2 = 0.5 \times 0.5 = 0.25

So, when x = 2, $f^{-1}(x) = 0.25$.

By calculating these values, we've successfully found the corresponding y-values for each x-value in our table. This step-by-step approach ensures we understand each calculation and how it contributes to completing the table. Now that we have all the values, let's put them together and see our completed table!

Completing the Table

Alright, we've done the hard work of calculating all the values. Now, it's time to put them together and complete the table for the inverse function $f^{-1}(x) = 0.5^x$. Remember, we calculated the values of $f^{-1}(x)$ for $x = -2, -1, 0, 1$, and $2$. Let's fill in the table:

x -2 -1 0 1 2
f⁻¹(x) 4 2 1 0.5 0.25

So, there you have it! We've successfully completed the table for the inverse function. This table now gives us a clear picture of how the inverse function behaves for these specific x-values. We can see how the function decreases as x increases, which is characteristic of an exponential decay function with a base between 0 and 1.

Completing tables like this is a fantastic way to visualize and understand functions. By plugging in different x-values and calculating the corresponding y-values, we get a tangible sense of the function's behavior. It's like creating a map of the function, showing us how it transforms inputs into outputs. This is especially useful when dealing with more complex functions where the behavior might not be immediately obvious.

Moreover, completing tables can help us identify patterns and trends in functions. In our case, we observed the decreasing nature of the exponential function as x increased. Such observations can lead to deeper insights about the function's properties and its applications in various fields. So, by completing this table, we've not only solved the problem but also gained a better understanding of inverse functions and their behavior.

Significance of the Results

Now that we've completed the table, let's take a moment to reflect on the significance of our results. We've found the values of the inverse function $f^{-1}(x) = 0.5^x$ for a range of x-values, and this gives us valuable insights into the nature of this function and its relationship with the original logarithmic function $f(x) = \log_{0.5} x$.

The values we calculated show the exponential decay of the function $f^{-1}(x)$. As x increases, the value of $f^{-1}(x)$ decreases, but it never actually reaches zero. This is a key characteristic of exponential decay functions. This behavior is directly linked to the properties of the original logarithmic function, which is defined only for positive x-values and approaches negative infinity as x approaches zero.

Understanding the behavior of inverse functions is crucial because it allows us to solve equations and model real-world phenomena. For instance, exponential functions (and their inverses, logarithmic functions) are used extensively in fields like finance (to model compound interest), physics (to describe radioactive decay), and biology (to model population growth or decay). The ability to work with these functions and understand their properties is a valuable skill in many areas.

Moreover, the relationship between a function and its inverse is a fundamental concept in mathematics. It highlights the idea of "undoing" an operation, which is a powerful tool in problem-solving. By understanding how a function transforms inputs into outputs and how its inverse reverses that transformation, we can gain a deeper understanding of mathematical relationships and apply them in various contexts.

In summary, the results we obtained by completing the table are not just numbers; they represent a deeper understanding of inverse functions, exponential decay, and the interconnectedness of mathematical concepts. This knowledge empowers us to tackle more complex problems and apply these principles in real-world situations.

Conclusion

So, there you have it, guys! We've successfully tackled the problem of completing the table for the inverse function $f^{-1}(x) = 0.5^x$. We started by understanding what inverse functions are, then we calculated the values for each x in the table, and finally, we reflected on the significance of our results. We've not only solved a math problem but also gained a deeper understanding of inverse functions and their applications.

Remember, math isn't just about numbers and equations; it's about understanding the relationships and patterns that govern the world around us. By working through problems like this, we sharpen our problem-solving skills and build a solid foundation for future mathematical explorations. Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!