Consider The Function:$\[ F(x) = 2 \frac{1}{2} - 3 \frac{1}{3} X \\]What Is The \[$ X \$\]-intercept Of \[$ F^{-1}(x) \$\]?A. \[$\left(-3 \frac{2}{3}, 0\right)\$\] B. \[$\left(-\frac{3}{10}, 0\right)\$\] C.

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Understanding the Problem

To find the x-intercept of the inverse function f−1(x)f^{-1}(x), we first need to understand the concept of an inverse function and how it relates to the original function. The inverse function f−1(x)f^{-1}(x) is a function that undoes the action of the original function f(x)f(x). In other words, if f(x)f(x) takes an input xx and produces an output yy, then f−1(x)f^{-1}(x) takes the input yy and produces the output xx.

The Original Function

The original function is given by the equation:

f(x)=212−313xf(x) = 2 \frac{1}{2} - 3 \frac{1}{3} x

This is a linear function, and we can rewrite it in the slope-intercept form as:

f(x)=52−103xf(x) = \frac{5}{2} - \frac{10}{3} x

Finding the Inverse Function

To find the inverse function, we need to swap the x and y variables and then solve for y. Let's start by writing the original function as:

y=52−103xy = \frac{5}{2} - \frac{10}{3} x

Now, we can swap the x and y variables to get:

x=52−103yx = \frac{5}{2} - \frac{10}{3} y

Next, we can solve for y by isolating it on one side of the equation. We can start by subtracting 52\frac{5}{2} from both sides to get:

x−52=−103yx - \frac{5}{2} = -\frac{10}{3} y

Now, we can multiply both sides by −310-\frac{3}{10} to get:

310(x−52)=y\frac{3}{10} (x - \frac{5}{2}) = y

Simplifying the left-hand side, we get:

310x−1520=y\frac{3}{10} x - \frac{15}{20} = y

Combining the fractions on the left-hand side, we get:

310x−34=y\frac{3}{10} x - \frac{3}{4} = y

Now, we can add 34\frac{3}{4} to both sides to get:

310x=y+34\frac{3}{10} x = y + \frac{3}{4}

Finally, we can multiply both sides by 103\frac{10}{3} to get:

x=103(y+34)x = \frac{10}{3} (y + \frac{3}{4})

Simplifying the right-hand side, we get:

x=103y+5x = \frac{10}{3} y + 5

The Inverse Function

The inverse function is given by the equation:

f−1(x)=103x+5f^{-1}(x) = \frac{10}{3} x + 5

Finding the x-Intercept

To find the x-intercept of the inverse function, we need to set f−1(x)=0f^{-1}(x) = 0 and solve for x. Let's start by writing the equation:

103x+5=0\frac{10}{3} x + 5 = 0

Subtracting 5 from both sides, we get:

103x=−5\frac{10}{3} x = -5

Multiplying both sides by 310\frac{3}{10}, we get:

x=−1510x = -\frac{15}{10}

Simplifying the fraction, we get:

x=−32x = -\frac{3}{2}

Conclusion

The x-intercept of the inverse function f−1(x)f^{-1}(x) is given by the point (−32,0)\left(-\frac{3}{2}, 0\right). This means that when the input to the inverse function is 0, the output is −32-\frac{3}{2}.

Answer

The correct answer is:

A. (−32,0)\left(-\frac{3}{2}, 0\right)

Understanding the Problem

In the previous article, we discussed how to find the x-intercept of the inverse function f−1(x)f^{-1}(x). In this article, we will answer some common questions related to this topic.

Q: What is the x-intercept of the inverse function f−1(x)f^{-1}(x)?

A: The x-intercept of the inverse function f−1(x)f^{-1}(x) is the point where the input to the inverse function is 0. In other words, it is the point where f−1(x)=0f^{-1}(x) = 0.

Q: How do I find the x-intercept of the inverse function f−1(x)f^{-1}(x)?

A: To find the x-intercept of the inverse function f−1(x)f^{-1}(x), you need to set f−1(x)=0f^{-1}(x) = 0 and solve for x. This involves swapping the x and y variables in the original function, solving for y, and then substituting the expression for y back into the original function.

Q: What is the difference between the original function and the inverse function?

A: The original function and the inverse function are related but distinct. The original function takes an input x and produces an output y, while the inverse function takes an input y and produces an output x.

Q: How do I know if the inverse function exists?

A: The inverse function exists if and only if the original function is one-to-one, meaning that each output value corresponds to exactly one input value. In other words, the original function must be either strictly increasing or strictly decreasing.

Q: Can I find the x-intercept of the inverse function f−1(x)f^{-1}(x) if the original function is not one-to-one?

A: No, you cannot find the x-intercept of the inverse function f−1(x)f^{-1}(x) if the original function is not one-to-one. In this case, the inverse function does not exist, and therefore, the x-intercept cannot be found.

Q: What is the significance of the x-intercept of the inverse function f−1(x)f^{-1}(x)?

A: The x-intercept of the inverse function f−1(x)f^{-1}(x) is significant because it represents the point where the input to the inverse function is 0. This point is important because it provides information about the behavior of the inverse function.

Q: Can I use the x-intercept of the inverse function f−1(x)f^{-1}(x) to find the x-intercept of the original function?

A: No, you cannot use the x-intercept of the inverse function f−1(x)f^{-1}(x) to find the x-intercept of the original function. The x-intercept of the original function is a different point, and it cannot be found using the x-intercept of the inverse function.

Conclusion

In this article, we answered some common questions related to finding the x-intercept of the inverse function f−1(x)f^{-1}(x). We hope that this article has provided you with a better understanding of this topic and has helped you to answer any questions you may have had.

Additional Resources

If you are interested in learning more about finding the x-intercept of the inverse function f−1(x)f^{-1}(x), we recommend checking out the following resources:

We hope that these resources are helpful to you.