Find The Inverse Function In Slope-intercept Form { (mx + B)$} : : : { F(x) = -\frac{2}{3}x + 14 \} ${ F^{-1}(x) = \square }$

Understanding the Slope-Intercept Form

The slope-intercept form of a linear equation is given by the formula $(mx + b)$, where $m$ represents the slope of the line and $b$ represents the y-intercept. This form is useful for graphing and analyzing linear equations. However, when dealing with inverse functions, we need to find the inverse of the given function in slope-intercept form.

The Given Function in Slope-Intercept Form

The given function in slope-intercept form is $f(x) = -\frac{2}{3}x + 14$. This function has a slope of $-\frac{2}{3}$ and a y-intercept of $14$. To find the inverse function, we need to swap the x and y variables and then solve for y.

Swapping the x and y Variables

To find the inverse function, we start by swapping the x and y variables in the given function. This gives us $x = -\frac{2}{3}y + 14$. Now, we need to solve for y.

Solving for y

To solve for y, we need to isolate y on one side of the equation. We can do this by subtracting $14$ from both sides of the equation, which gives us $x - 14 = -\frac{2}{3}y$. Next, we can multiply both sides of the equation by $-\frac{3}{2}$ to get rid of the fraction. This gives us $-\frac{3}{2}(x - 14) = y$.

Simplifying the Equation

Now, we can simplify the equation by distributing the $-\frac{3}{2}$ to the terms inside the parentheses. This gives us $-\frac{3}{2}x + 21 = y$. Now, we can rewrite the equation in slope-intercept form by isolating y on one side of the equation.

Rewriting the Equation in Slope-Intercept Form

The equation $-\frac{3}{2}x + 21 = y$ is already in slope-intercept form, where the slope is $-\frac{3}{2}$ and the y-intercept is $21$. However, we need to express the equation in terms of $f^{-1}(x)$, which is the inverse function of $f(x)$.

Expressing the Inverse Function

To express the inverse function, we can rewrite the equation as $f^{-1}(x) = -\frac{3}{2}x + 21$. This is the inverse function of the given function $f(x) = -\frac{2}{3}x + 14$.

Conclusion

In this article, we have learned how to find the inverse function in slope-intercept form. We started by understanding the slope-intercept form of a linear equation and then applied this concept to find the inverse function of a given function. We swapped the x and y variables, solved for y, and simplified the equation to express the inverse function in slope-intercept form. The inverse function of the given function $f(x) = -\frac{2}{3}x + 14$ is $f^{-1}(x) = -\frac{3}{2}x + 21$.

Example Problems

Problem 1

Find the inverse function of the given function $f(x) = \frac{1}{2}x - 5$.

Solution

To find the inverse function, we start by swapping the x and y variables in the given function. This gives us $x = \frac{1}{2}y - 5$. Now, we need to solve for y.

To solve for y, we can add $5$ to both sides of the equation, which gives us $x + 5 = \frac{1}{2}y$. Next, we can multiply both sides of the equation by $2$ to get rid of the fraction. This gives us $2(x + 5) = y$.

Now, we can simplify the equation by distributing the $2$ to the terms inside the parentheses. This gives us $2x + 10 = y$. Now, we can rewrite the equation in slope-intercept form by isolating y on one side of the equation.

The equation $2x + 10 = y$ is already in slope-intercept form, where the slope is $2$ and the y-intercept is $10$. However, we need to express the equation in terms of $f^{-1}(x)$, which is the inverse function of $f(x)$.

To express the inverse function, we can rewrite the equation as $f^{-1}(x) = 2x + 10$. This is the inverse function of the given function $f(x) = \frac{1}{2}x - 5$.

Problem 2

Find the inverse function of the given function $f(x) = -\frac{1}{4}x + 12$.

Solution

To find the inverse function, we start by swapping the x and y variables in the given function. This gives us $x = -\frac{1}{4}y + 12$. Now, we need to solve for y.

To solve for y, we can subtract $12$ from both sides of the equation, which gives us $x - 12 = -\frac{1}{4}y$. Next, we can multiply both sides of the equation by $-4$ to get rid of the fraction. This gives us $-4(x - 12) = y$.

Now, we can simplify the equation by distributing the $-4$ to the terms inside the parentheses. This gives us $-4x + 48 = y$. Now, we can rewrite the equation in slope-intercept form by isolating y on one side of the equation.

The equation $-4x + 48 = y$ is already in slope-intercept form, where the slope is $-4$ and the y-intercept is $48$. However, we need to express the equation in terms of $f^{-1}(x)$, which is the inverse function of $f(x)$.

To express the inverse function, we can rewrite the equation as $f^{-1}(x) = -4x + 48$. This is the inverse function of the given function $f(x) = -\frac{1}{4}x + 12$.

Tips and Tricks

• When finding the inverse function, make sure to swap the x and y variables in the given function.
• Solve for y by isolating y on one side of the equation.
• Simplify the equation by distributing the coefficients to the terms inside the parentheses.
• Rewrite the equation in slope-intercept form by isolating y on one side of the equation.
• Express the inverse function in terms of $f^{-1}(x)$, which is the inverse function of $f(x)$.

Conclusion

In this article, we have learned how to find the inverse function in slope-intercept form. We started by understanding the slope-intercept form of a linear equation and then applied this concept to find the inverse function of a given function. We swapped the x and y variables, solved for y, and simplified the equation to express the inverse function in slope-intercept form. The inverse function of the given function $f(x) = -\frac{2}{3}x + 14$ is $f^{-1}(x) = -\frac{3}{2}x + 21$. We also provided example problems and tips and tricks to help you understand the concept better.

Q: What is the slope-intercept form of a linear equation?

A: The slope-intercept form of a linear equation is given by the formula $(mx + b)$, where $m$ represents the slope of the line and $b$ represents the y-intercept.

Q: How do I find the inverse function of a given function in slope-intercept form?

A: To find the inverse function, you need to swap the x and y variables in the given function and then solve for y. You can do this by adding or subtracting the constant term to both sides of the equation, and then multiplying or dividing both sides of the equation by the coefficient of the x term.

Q: What is the inverse function of the given function $f(x) = -\frac{2}{3}x + 14$?

A: The inverse function of the given function $f(x) = -\frac{2}{3}x + 14$ is $f^{-1}(x) = -\frac{3}{2}x + 21$.

Q: How do I express the inverse function in terms of $f^{-1}(x)$?

A: To express the inverse function in terms of $f^{-1}(x)$, you need to rewrite the equation in slope-intercept form by isolating y on one side of the equation.

Q: What are some common mistakes to avoid when finding the inverse function?

A: Some common mistakes to avoid when finding the inverse function include:

• Swapping the x and y variables incorrectly
• Solving for y incorrectly
• Simplifying the equation incorrectly
• Expressing the inverse function incorrectly

Q: How do I check if the inverse function is correct?

A: To check if the inverse function is correct, you can plug in a value of x into the inverse function and see if it gives you the correct value of y.

Q: What are some real-world applications of finding the inverse function?

A: Some real-world applications of finding the inverse function include:

• Modeling population growth and decline
• Analyzing the relationship between two variables
• Finding the optimal solution to a problem

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

A: Yes, you can use a calculator to find the inverse function. However, it's always a good idea to double-check your work by plugging in a value of x into the inverse function.

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

A: To find the inverse function of a quadratic equation, you need to first rewrite the equation in vertex form. Then, you can swap the x and y variables and solve for y.

Q: Can I find the inverse function of a non-linear equation?

A: No, you cannot find the inverse function of a non-linear equation. The inverse function only exists for linear equations.

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

A: To find the inverse function of a piecewise function, you need to find the inverse function for each individual piece of the function and then combine them.

Q: Can I use the inverse function to solve a system of equations?

A: Yes, you can use the inverse function to solve a system of equations. However, it's always a good idea to check your work by plugging in a value of x into the inverse function.

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

A: To find the inverse function of a function with multiple variables, you need to first rewrite the equation in terms of one variable. Then, you can swap the x and y variables and solve for y.

Q: Can I use the inverse function to model real-world phenomena?

A: Yes, you can use the inverse function to model real-world phenomena. However, it's always a good idea to check your work by plugging in a value of x into the inverse function.

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

A: To find the inverse function of a function with a complex coefficient, you need to first rewrite the equation in terms of one variable. Then, you can swap the x and y variables and solve for y.

Q: Can I use the inverse function to solve a differential equation?

A: Yes, you can use the inverse function to solve a differential equation. However, it's always a good idea to check your work by plugging in a value of x into the inverse function.

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

A: To find the inverse function of a function with a trigonometric coefficient, you need to first rewrite the equation in terms of one variable. Then, you can swap the x and y variables and solve for y.

Q: Can I use the inverse function to model the behavior of a system?

A: Yes, you can use the inverse function to model the behavior of a system. However, it's always a good idea to check your work by plugging in a value of x into the inverse function.

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

A: To find the inverse function of a function with a logarithmic coefficient, you need to first rewrite the equation in terms of one variable. Then, you can swap the x and y variables and solve for y.

Q: Can I use the inverse function to solve a system of differential equations?

A: Yes, you can use the inverse function to solve a system of differential equations. However, it's always a good idea to check your work by plugging in a value of x into the inverse function.

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

A: To find the inverse function of a function with a polynomial coefficient, you need to first rewrite the equation in terms of one variable. Then, you can swap the x and y variables and solve for y.

Q: Can I use the inverse function to model the behavior of a population?

A: Yes, you can use the inverse function to model the behavior of a population. However, it's always a good idea to check your work by plugging in a value of x into the inverse function.

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

A: To find the inverse function of a function with a rational coefficient, you need to first rewrite the equation in terms of one variable. Then, you can swap the x and y variables and solve for y.

Q: Can I use the inverse function to solve a system of equations with multiple variables?

A: Yes, you can use the inverse function to solve a system of equations with multiple variables. However, it's always a good idea to check your work by plugging in a value of x into the inverse function.

Q: How do I find the inverse function of a function with a complex coefficient and multiple variables?

A: To find the inverse function of a function with a complex coefficient and multiple variables, you need to first rewrite the equation in terms of one variable. Then, you can swap the x and y variables and solve for y.

Q: Can I use the inverse function to model the behavior of a system with multiple variables?

A: Yes, you can use the inverse function to model the behavior of a system with multiple variables. However, it's always a good idea to check your work by plugging in a value of x into the inverse function.

Q: How do I find the inverse function of a function with a rational coefficient and multiple variables?

A: To find the inverse function of a function with a rational coefficient and multiple variables, you need to first rewrite the equation in terms of one variable. Then, you can swap the x and y variables and solve for y.

Q: Can I use the inverse function to solve a system of differential equations with multiple variables?

A: Yes, you can use the inverse function to solve a system of differential equations with multiple variables. However, it's always a good idea to check your work by plugging in a value of x into the inverse function.

Q: How do I find the inverse function of a function with a polynomial coefficient and multiple variables?

A: To find the inverse function of a function with a polynomial coefficient and multiple variables, you need to first rewrite the equation in terms of one variable. Then, you can swap the x and y variables and solve for y.

Q: Can I use the inverse function to model the behavior of a population with multiple variables?

A: Yes, you can use the inverse function to model the behavior of a population with multiple variables. However, it's always a good idea to check your work by plugging in a value of x into the inverse function.

Q: How do I find the inverse function of a function with a logarithmic coefficient and multiple variables?

A: To find the inverse function of a function with a logarithmic coefficient and multiple variables, you need to first rewrite the equation in terms of one variable. Then, you can swap the x and y variables and solve for y.

Q: Can I use the inverse function to solve a system of equations with complex coefficients and multiple variables?

A: Yes, you can use the inverse function to solve a system of equations with complex coefficients and multiple variables. However, it's always a good idea to check your work by plugging in a