Given The Function F ( X ) = 2 X − 6 F(x)=2x-6 F ( X ) = 2 X − 6 .(a) Find F − 1 F^{-1} F − 1 .(b) Graph F F F And F − 1 F^{-1} F − 1 In The Same Rectangular Coordinate System.(c) Use Interval Notation To Give The Domain And The Range Of F F F And

Understanding the Problem

Given the function $f(x) = 2x - 6$, we are tasked with finding its inverse, graphing both functions in the same rectangular coordinate system, and determining the domain and range of $f$ and $f^{-1}$ using interval notation.

Finding the Inverse of a Linear Function

To find the inverse of a linear function, we need to swap the roles of $x$ and $y$ and then solve for $y$. The original function is $f(x) = 2x - 6$. We start by writing $y = 2x - 6$.

Swapping the Roles of $x$ and $y$

We swap the roles of $x$ and $y$ to get $x = 2y - 6$.

Solving for $y$

Now, we solve for $y$ by isolating it on one side of the equation. We add $6$ to both sides to get $x + 6 = 2y$. Then, we divide both sides by $2$ to get $y = \frac{x + 6}{2}$.

Finding the Inverse Function

The inverse function is denoted as $f^{-1}(x)$. Therefore, we have $f^{-1}(x) = \frac{x + 6}{2}$.

Graphing $f$ and $f^{-1}$

To graph $f$ and $f^{-1}$ in the same rectangular coordinate system, we need to plot the points for both functions.

Graphing $f(x) = 2x - 6$

We start by finding the $y$-intercept of $f(x) = 2x - 6$. We substitute $x = 0$ into the equation to get $y = 2(0) - 6 = -6$. Therefore, the $y$-intercept is $(-6, 0)$.

Next, we find the $x$-intercept of $f(x) = 2x - 6$. We substitute $y = 0$ into the equation to get $0 = 2x - 6$. We add $6$ to both sides to get $6 = 2x$. Then, we divide both sides by $2$ to get $x = 3$. Therefore, the $x$-intercept is $(3, 0)$.

We plot the points $(-6, 0)$ and $(3, 0)$ and draw a line through them to get the graph of $f(x) = 2x - 6$.

Graphing $f^{-1}(x) = \frac{x + 6}{2}$

We start by finding the $y$-intercept of $f^{-1}(x) = \frac{x + 6}{2}$. We substitute $x = 0$ into the equation to get $y = \frac{0 + 6}{2} = 3$. Therefore, the $y$-intercept is $(0, 3)$.

Next, we find the $x$-intercept of $f^{-1}(x) = \frac{x + 6}{2}$. We substitute $y = 0$ into the equation to get $0 = \frac{x + 6}{2}$. We multiply both sides by $2$ to get $0 = x + 6$. Then, we subtract $6$ from both sides to get $-6 = x$. Therefore, the $x$-intercept is $(-6, 0)$.

We plot the points $(0, 3)$ and $(-6, 0)$ and draw a line through them to get the graph of $f^{-1}(x) = \frac{x + 6}{2}$.

Domain and Range of $f$ and $f^{-1}$

The domain of a function is the set of all possible input values for which the function is defined. The range of a function is the set of all possible output values for which the function is defined.

Domain and Range of $f(x) = 2x - 6$

The domain of $f(x) = 2x - 6$ is all real numbers, which can be written as $(-\infty, \infty)$. The range of $f(x) = 2x - 6$ is also all real numbers, which can be written as $(-\infty, \infty)$.

Domain and Range of $f^{-1}(x) = \frac{x + 6}{2}$

The domain of $f^{-1}(x) = \frac{x + 6}{2}$ is all real numbers, which can be written as $(-\infty, \infty)$. The range of $f^{-1}(x) = \frac{x + 6}{2}$ is also all real numbers, which can be written as $(-\infty, \infty)$.

Conclusion

In this article, we found the inverse of the linear function $f(x) = 2x - 6$, graphed both functions in the same rectangular coordinate system, and determined the domain and range of $f$ and $f^{-1}$ using interval notation. We showed that the inverse function is $f^{-1}(x) = \frac{x + 6}{2}$ and that the domain and range of both functions are all real numbers.

Understanding the Problem

In the previous article, we found the inverse of the linear function $f(x) = 2x - 6$, graphed both functions in the same rectangular coordinate system, and determined the domain and range of $f$ and $f^{-1}$ using interval notation. In this article, we will answer some frequently asked questions about finding the inverse of a linear function and graphing.

Q: What is the inverse of a linear function?

A: The inverse of a linear function is a function that undoes the action of the original function. In other words, if we apply the original function to a value, and then apply the inverse function to the result, we should get back the original value.

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

A: To find the inverse of a linear function, we need to swap the roles of $x$ and $y$ and then solve for $y$. We start by writing the original function as $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept. We then swap the roles of $x$ and $y$ to get $x = my + b$. We solve for $y$ by isolating it on one side of the equation.

Q: What is the graph of the inverse of a linear function?

A: The graph of the inverse of a linear function is a reflection of the graph of the original function across the line $y = x$. This means that if we have a point $(x, y)$ on the graph of the original function, the corresponding point on the graph of the inverse function is $(y, x)$.

Q: How do I determine the domain and range of a linear function and its inverse?

A: The domain of a linear function is all real numbers, which can be written as $(-\infty, \infty)$. The range of a linear function is also all real numbers, which can be written as $(-\infty, \infty)$. The domain and range of the inverse function are the same as the domain and range of the original function.

Q: Can I use a graphing calculator to find the inverse of a linear function?

A: Yes, you can use a graphing calculator to find the inverse of a linear function. You can graph the original function and then use the calculator's inverse function feature to find the inverse function.

Q: How do I check if my answer is correct?

A: To check if your answer is correct, you can graph both the original function and the inverse function on the same coordinate plane. If the graphs are reflections of each other across the line $y = x$, then your answer is correct.

Q: What if I have a non-linear function?

A: If you have a non-linear function, you cannot find its inverse using the same method as for a linear function. You will need to use a different method, such as substitution or elimination, to find the inverse.

Conclusion

In this article, we answered some frequently asked questions about finding the inverse of a linear function and graphing. We showed that the inverse of a linear function is a function that undoes the action of the original function, and that the graph of the inverse function is a reflection of the graph of the original function across the line $y = x$. We also discussed how to determine the domain and range of a linear function and its inverse, and how to use a graphing calculator to find the inverse of a linear function.