Question 22.1 Determine The Inverses Of The Following Functions:2.1.1 $f(x)=\frac{2}{3} X$2.1.2 $g(x)=-3x-9$1.4 Determine The Point(s) At Which F − 1 F^{-1} F − 1 And G − 1 G^{-1} G − 1 Will Intersect.Given The Function

Introduction

In mathematics, the concept of inverse functions is crucial in understanding the relationship between two functions. Given a function, its inverse function undoes the action of the original function, and vice versa. In this article, we will explore the process of finding the inverses of two given functions, $f(x)=\frac{2}{3} x$ and $g(x)=-3x-9$, and then determine the point(s) at which their inverses, $f^{-1}$ and $g^{-1}$, will intersect.

Finding the Inverse of $f(x)=\frac{2}{3} x$

To find the inverse of a function, we need to swap the roles of $x$ and $y$ and then solve for $y$. Let's start with the function $f(x)=\frac{2}{3} x$. We can rewrite this function as $y=\frac{2}{3} x$.

Step 1: Swap the roles of $x$ and $y$

We swap the roles of $x$ and $y$ to get $x=\frac{2}{3} y$.

Step 2: Solve for $y$

To solve for $y$, we need to isolate $y$ on one side of the equation. We can do this by multiplying both sides of the equation by $\frac{3}{2}$.

x = (2/3) * y
x = (3/2) * x
y = (3/2) * x

Step 3: Write the inverse function

The inverse function of $f(x)=\frac{2}{3} x$ is $f^{-1}(x)=\frac{3}{2} x$.

Finding the Inverse of $g(x)=-3x-9$

To find the inverse of a function, we need to swap the roles of $x$ and $y$ and then solve for $y$. Let's start with the function $g(x)=-3x-9$. We can rewrite this function as $y=-3x-9$.

Step 1: Swap the roles of $x$ and $y$

We swap the roles of $x$ and $y$ to get $x=-3y-9$.

Step 2: Solve for $y$

To solve for $y$, we need to isolate $y$ on one side of the equation. We can do this by adding $9$ to both sides of the equation and then dividing both sides by $-3$.

x = -3y - 9
x + 9 = -3y
(x + 9) / -3 = y
y = (-x - 9) / 3

Step 3: Write the inverse function

The inverse function of $g(x)=-3x-9$ is $g^{-1}(x)=\frac{-x-9}{3}$.

Finding the Intersection Point(s) of $f^{-1}$ and $g^{-1}$

To find the intersection point(s) of $f^{-1}$ and $g^{-1}$, we need to set the two functions equal to each other and solve for $x$. Let's set $f^{-1}(x)=g^{-1}(x)$ and solve for $x$.

f^{-1}(x) = g^{-1}(x)
(3/2)x = (-x - 9) / 3
(9/2)x = -x - 9
(9/2)x + x = -9
(11/2)x = -9
x = -18/11

The intersection point of $f^{-1}$ and $g^{-1}$ is $x=-\frac{18}{11}$.

Conclusion

Introduction

In our previous article, we explored the process of finding the inverses of two given functions, $f(x)=\frac{2}{3} x$ and $g(x)=-3x-9$, and then determined the point(s) at which their inverses, $f^{-1}$ and $g^{-1}$, will intersect. In this article, we will answer some frequently asked questions related to finding inverses and intersection points of functions.

Q: What is the purpose of finding the inverse of a function?

A: The purpose of finding the inverse of a function is to understand the relationship between two functions. The inverse function undoes the action of the original function, and vice versa. This is useful in many areas of mathematics, such as solving equations, graphing functions, and analyzing the behavior of functions.

Q: How do I find the inverse of a function?

A: To find the inverse of a function, you need to swap the roles of $x$ and $y$ and then solve for $y$. This involves rewriting the function in terms of $x$ and $y$, swapping the variables, and then solving for $y$.

Q: What is the difference between a function and its inverse?

A: A function and its inverse are two functions that are related to each other. The inverse function undoes the action of the original function, and vice versa. For example, if $f(x)=2x$, then the inverse function $f^{-1}(x)=\frac{x}{2}$.

Q: How do I find the intersection point(s) of two functions?

A: To find the intersection point(s) of two functions, you need to set the two functions equal to each other and solve for $x$. This involves rewriting the two functions in terms of $x$ and $y$, setting them equal to each other, and then solving for $x$.

Q: What is the significance of the intersection point(s) of two functions?

A: The intersection point(s) of two functions is the point(s) at which the two functions intersect. This is useful in many areas of mathematics, such as solving equations, graphing functions, and analyzing the behavior of functions.

Q: Can two functions have more than one intersection point?

A: Yes, two functions can have more than one intersection point. This occurs when the two functions intersect at multiple points.

Q: How do I determine the number of intersection points of two functions?

A: To determine the number of intersection points of two functions, you need to analyze the graphs of the two functions and look for the points at which they intersect. You can also use algebraic methods, such as solving the equation $f(x)=g(x)$, to determine the number of intersection points.

Q: What is the relationship between the graph of a function and its inverse?

A: The graph of a function and its inverse are related in the sense that the graph of the inverse function is the reflection of the graph of the original function across the line $y=x$. This means that if the graph of the original function is a curve, the graph of the inverse function will be a curve that is reflected across the line $y=x$.

Conclusion

In this article, we answered some frequently asked questions related to finding inverses and intersection points of functions. We hope that this article has provided you with a better understanding of the concepts of inverse functions and intersection points. If you have any further questions, please don't hesitate to ask.