Evaluating the Inverse Function

Oct 13, 2025 by windailys.com 58 views

Introduction

In mathematics, the inverse of a function is a function that reverses the operation of the original function. In this article, we will explore the inverse of the logarithmic function and find the values of a, b, and c that complete the table for the inverse function.

The Logarithmic Function

The logarithmic function is defined as $f(x) = \log_{a}(x)$ , where a is the base of the logarithm. The inverse of the logarithmic function is defined as $f^{-1}(x) = a^x$ . In this case, the base of the logarithm is 0.5, so the inverse function is $f^{-1}(x) = 0.5^x$ .

Finding the Values of a, b, and c

To find the values of a, b, and c that complete the table for the inverse function, we need to evaluate the inverse function at different values of x. The table is given as:

x	-2	-1	0	1	2
f(x)	a	b	c	d	e

We need to find the values of a, b, and c.

Evaluating the Inverse Function

To evaluate the inverse function, we substitute the values of x into the equation $f^{-1}(x) = 0.5^x$ . Let's start with x = -2.

$f^{-1}(-2) = 0.5^{-2} = \frac{1}{0.5^2} = \frac{1}{0.25} = 4$

So, the value of a is 4.

Evaluating the Inverse Function at x = -1

Now, let's evaluate the inverse function at x = -1.

$f^{-1}(-1) = 0.5^{-1} = \frac{1}{0.5^1} = \frac{1}{0.5} = 2$

So, the value of b is 2.

Evaluating the Inverse Function at x = 0

Next, let's evaluate the inverse function at x = 0.

$f^{-1}(0) = 0.5^0 = 1$

So, the value of c is 1.

Conclusion

In this article, we found the values of a, b, and c that complete the table for the inverse function. The values are a = 4, b = 2, and c = 1.

The Complete Table

Here is the complete table:

x	-2	-1	0	1	2
f(x)	4	2	1	d	e

We still need to find the values of d and e.

Finding the Values of d and e

To find the values of d and e, we need to evaluate the inverse function at x = 1 and x = 2.

Evaluating the Inverse Function at x = 1

Let's evaluate the inverse function at x = 1.

$f^{-1}(1) = 0.5^1 = 0.5$

So, the value of d is 0.5.

Evaluating the Inverse Function at x = 2

Next, let's evaluate the inverse function at x = 2.

$f^{-1}(2) = 0.5^2 = 0.25$

So, the value of e is 0.25.

The Complete Table

Here is the complete table:

x	-2	-1	0	1	2
f(x)	4	2	1	0.5	0.25

Conclusion

In this article, we found the values of a, b, c, d, and e that complete the table for the inverse function. The values are a = 4, b = 2, c = 1, d = 0.5, and e = 0.25.

The Inverse of the Logarithmic Function

The inverse of the logarithmic function is a powerful tool in mathematics. It allows us to reverse the operation of the original function and find the input value that corresponds to a given output value. In this article, we found the values of a, b, c, d, and e that complete the table for the inverse function.

Applications of the Inverse of the Logarithmic Function

The inverse of the logarithmic function has many applications in mathematics and science. It is used in calculus, algebra, and number theory, among other areas. It is also used in real-world applications such as finance, engineering, and computer science.

Conclusion

Q: What is the inverse of the logarithmic function?

A: The inverse of the logarithmic function is a function that reverses the operation of the original function. In this case, the inverse function is $f^{-1}(x) = 0.5^x$ .

Q: How do you find the inverse of a logarithmic function?

A: To find the inverse of a logarithmic function, you need to swap the x and y variables and then solve for y. In this case, we started with the equation $f(x) = \log_{0.5}(x)$ and swapped the x and y variables to get $x = \log_{0.5}(y)$ . Then, we solved for y to get $y = 0.5^x$ .

Q: What are the values of a, b, and c in the table for the inverse function?

A: The values of a, b, and c in the table for the inverse function are a = 4, b = 2, and c = 1.

Q: How do you evaluate the inverse function at different values of x?

A: To evaluate the inverse function at different values of x, you substitute the values of x into the equation $f^{-1}(x) = 0.5^x$ . For example, to evaluate the inverse function at x = -2, you substitute x = -2 into the equation to get $f^{-1}(-2) = 0.5^{-2} = \frac{1}{0.5^2} = \frac{1}{0.25} = 4$ .

Q: What are the values of d and e in the table for the inverse function?

A: The values of d and e in the table for the inverse function are d = 0.5 and e = 0.25.

Q: What are some applications of the inverse of the logarithmic function?

A: The inverse of the logarithmic function has many applications in mathematics and science. It is used in calculus, algebra, and number theory, among other areas. It is also used in real-world applications such as finance, engineering, and computer science.

Q: How do you use the inverse of the logarithmic function in real-world applications?

A: The inverse of the logarithmic function is used in real-world applications such as finance, engineering, and computer science. For example, it is used to calculate the interest rate on a loan, to determine the amount of money that will be needed to pay off a loan, and to calculate the probability of a certain event occurring.

Q: What are some common mistakes to avoid when working with the inverse of the logarithmic function?

A: Some common mistakes to avoid when working with the inverse of the logarithmic function include:

Not swapping the x and y variables when finding the inverse of a function
Not solving for y when finding the inverse of a function
Not evaluating the inverse function at different values of x
Not using the correct values of a, b, and c in the table for the inverse function

Q: How do you troubleshoot common mistakes when working with the inverse of the logarithmic function?

A: To troubleshoot common mistakes when working with the inverse of the logarithmic function, you can:

Check your work to make sure that you swapped the x and y variables and solved for y
Check your work to make sure that you evaluated the inverse function at different values of x
Check your work to make sure that you used the correct values of a, b, and c in the table for the inverse function
Ask for help from a teacher or tutor if you are unsure about how to work with the inverse of the logarithmic function.

Conclusion

In conclusion, the inverse of the logarithmic function is a powerful tool in mathematics. It allows us to reverse the operation of the original function and find the input value that corresponds to a given output value. In this article, we answered some common questions about the inverse of the logarithmic function and provided some tips for troubleshooting common mistakes.