Inverse of One-to-One Function | Grade 11- General Mathematics

Summary

This educational video tutorial teaches viewers how to find the inverse of one-to-one functions. It covers three examples: solving for the inverse of a linear function (f(x) = 5x - 8), a rational function (f(x) = (x - 4)/(2x - 7)), and a cubic function (f(x) = 2x^3 - 5). The process involves changing f(x) to y, swapping x and y, solving for y, and simplifying. The video concludes with the inverse functions for each example, encouraging viewers to like and subscribe for more educational content.

Takeaways

• 📕 To find the inverse of a one-to-one function, you first replace f(x) with y.
• 📝 Interchange the variables x and y to reflect the inverse relationship.
• 💵 Solve for y to derive the inverse function.
• 💲 For a linear function like f(x) = 5x - 8, isolate y by transposing terms and dividing by the coefficient.
• 💳 The inverse of f(x) = 5x - 8 is y = (x + 8)/5.
• 📘 For rational expressions, cross-multiply to eliminate the fraction.
• 💴 Group terms with y on one side and terms without y on the other side to isolate y.
• 💹 The inverse of f(x) = (x - 4)/(2x - 7) is y = (7x - 4)/(2x - 1).
• 📗 For functions involving exponents, such as f(x) = 2x^3 - 5, solve for y by transposing and dividing to isolate the cube root.
• 💱 The inverse of f(x) = 2x^3 - 5 is y = √x + 5/2∛.
• 📝 The video provides a step-by-step guide on how to find the inverse of different types of functions, including linear, rational, and cubic functions.

Q & A

What is the first step in finding the inverse of a one-to-one function?

The first step is to change f(x) to y, so that y equals the function expression.

How do you interchange variables to find the inverse function?

You interchange x and y variables in the equation to express x in terms of y.

What is the inverse function of f(x) = 5x - 8?

The inverse function of f(x) = 5x - 8 is x + 8/5.

How do you solve for y in the equation x = 5y - 8 after interchanging variables?

You transpose -8 to the left side to get +8 and then divide both sides by 5 to isolate y.

What is the process for solving the inverse of a rational function like f(x) = (x - 4)/(2x - 7)?

You interchange x and y, cross multiply to eliminate the fraction, and then group terms with y on one side and terms without y on the other side.

How do you handle the rational expression after interchanging x and y in f(x) = (x - 4)/(2x - 7)?

You cross multiply to get rid of the fraction, then rearrange terms to isolate y.

What is the inverse function of f(x) = (x - 4)/(2x - 7)?

The inverse function of f(x) = (x - 4)/(2x - 7) is 7x - 4/(2x - 1).

How do you solve for y in the equation x = y^3 - 5 after interchanging variables?

You transpose -5 to the other side to get +5, then divide both sides by 2, and finally take the cube root of both sides.

What is the inverse function of f(x) = 2x^3 - 5?

The inverse function of f(x) = 2x^3 - 5 is the cube root of (x + 5)/2.

Why is it necessary to cube root both sides after isolating y in the equation x = y^3 - 5?

You cube root both sides to eliminate the power of 3 on the right side and isolate y.

What does it mean when the inverse function is represented with a negative exponent on y?

A negative exponent on y indicates that the function is the inverse of the original function.

Outlines

00:00

📘 Finding the Inverse of a Linear Function

The paragraph explains how to find the inverse of a one-to-one function starting with a linear function f(x) = 5x - 8. The process involves changing f(x) to y, swapping x and y to get x = 5y - 8, and then solving for y. By transposing -8 to the other side and dividing by 5, the inverse function is derived as y = (x + 8)/5.

📗 Inverting a Rational Function

This section details the inversion of a rational function f(x) = (x - 4) / (2x - 7). The method involves rewriting y = f(x), swapping x and y, and then cross-multiplying to eliminate the fraction. The equation is simplified by moving terms involving y to one side and solving for y, resulting in the inverse function y = (7x - 4) / (2x - 1).

📙 Inverting a Cubic Function

The final example demonstrates inverting a cubic function f(x) = 2x^3 - 5. The process includes rewriting the function with y, swapping x and y, and then solving for y by transposing terms and dividing by the coefficient of y. The final step involves taking the cube root of both sides to isolate y, resulting in the inverse function y = ∛(x + 5)/2.

Keywords

💡Inverse Function

An inverse function is a function that 'undoes' the action of another function. In the context of the video, it's about finding a function that, when applied to the output of the original function, returns the original input. This is a central theme of the video, as it demonstrates how to find the inverse of various functions.

💡One-to-One Function

A one-to-one function is a type of function where each output is produced by exactly one input. This property is essential for a function to have an inverse because it ensures that reversing the function will yield a unique result. The video focuses on inverting one-to-one functions.

💡Interchange Variables

Interchanging variables is a step in finding the inverse of a function where you swap the roles of the input and output variables. In the video, this is done by setting y equals the function of x and then solving for x in terms of y, which is a crucial step in the inversion process.

💡Transposing

Transposing in the context of the video refers to moving terms from one side of an equation to another to isolate the variable you are solving for. This is a common algebraic technique used in the examples to simplify equations and solve for y in terms of x.

💡Solving for y

Solving for y is the process of manipulating the equation to express y in terms of x, which is necessary to find the inverse function. The video provides examples of how to algebraically manipulate equations to achieve this.

💡Cross Multiply

Cross multiplying is a technique used when dealing with rational expressions to eliminate the fractions by multiplying both sides of the equation by the product of the denominators. The video uses this method in the second example to simplify the equation and solve for y.

💡Grouping Terms

Grouping terms is an algebraic method where similar terms are collected on one side of the equation to simplify it. In the video, this is done to isolate terms with the variable y on one side and constants on the other, facilitating the solution for y.

💡Dividing by a Coefficient

Dividing by a coefficient is a step taken to isolate the variable by getting rid of the multiplier in front of it. The video demonstrates this by dividing both sides of the equation by the coefficient of y to solve for y.

💡Cube Root

A cube root is the inverse operation of cubing a number, which means finding a number that, when cubed, gives the original number. In the video's last example, taking the cube root is part of the process to find the inverse of a cubic function.

💡Cubing

Cubing a number means raising it to the power of three. In the context of the video, cubing is used in the original function, and the inverse operation (cube root) is needed to find the inverse function.

💡Rational Expression

A rational expression is a fraction where the numerator and/or the denominator are polynomials. The video's second example involves a rational expression, and the process of finding its inverse requires special handling to deal with the fractions.

Highlights

Introduction to solving the inverse of one-to-one functions

Example 1: Inverse of f(x) = 5x - 8

Step 1: Change f(x) to y = 5x - 8

Step 2: Interchange Y and X variables

Step 3: Solve for y by isolating it

Inverse of f(x) = 5x - 8 is x + 8/5

Example 2: Inverse of f(x) = (x - 4)/(2x - 7)

Step 1: Change f(x) to y and interchange variables

Step 2: Cross multiply to eliminate the fraction

Step 3: Group terms with Y and without Y

Step 4: Solve for y by isolating it

Inverse of f(x) = (x - 4)/(2x - 7) is 7x - 4/(2x - 1)

Example 3: Inverse of f(x) = 2x^3 - 5

Step 1: Change f(x) to y and interchange variables

Step 2: Solve for y by isolating it

Step 3: Eliminate the power 3 by taking the cube root

Inverse of f(x) = 2x^3 - 5 is the cube root of (x + 5)/2

Summary of the method to find inverse functions

Encouragement to like and subscribe for more educational content

Transcripts

in this video I will show you how to

solve the inverse of one-to-one function

our first example is f of x equals 5x

minus 8. first thing to do is to change

f of x to y y equals 5x minus 8.

next interchange Y and X variables x

equals 5y minus eight

then solved for y first let us transpose

negative 8 to the left side which will

become positive eight next to eliminate

five on the right side divide both sides

with five

five on the right side will be canceled

this will be X Plus 8 over 5 equals y or

Y equals X plus eight over five

therefore the inverse of f of x equals

5x minus 8 is X Plus 8 over 5.

our second example is f of x equals x

minus 4 over 2x minus seven

first step is to change f of x to y y

equals x minus four over two x minus

seven then interchange X and Y variables

x equals y minus 4 over 2y minus 7.

since we have a rational expression let

us cross multiply x times 2y minus 7

equals y minus 4 times 1 x times 2y is

2xy x times negative 7 is negative 7x

equals 1 times Y is y 1 times negative 4

is negative four

now we have 2xy minus 7x equals y minus

4. let us group all terms with Y

variable on one side and all terms with

no y variable on the other side

copy to X Y transpose y

transpose negative 7x which will be

positive 7X

copy negative 4.

now 2xy and negative y have a common

factor y a y times the quantity of 2xy

divided by Y is 2x minus y divided by Y

is 1 equals 7x minus four next let us

divide both sides with 2x minus 1. this

will be canceled

y equals 7x minus 4 over 2x minus one

let us Put negative one on the power of

Y as a representation that this is the

inverse function

therefore the inverse of f of x equals x

minus 4 over 2x minus 7 is 7x minus 4

over 2 x minus 1.

last example f of x equals 2x cubed

minus 5. First Step change f of x to y

interchange X and Y variables x equals

to Y cubed minus five

solve for y

let us transpose negative 5 and it will

be positive 5. next to eliminate 2

divide both sides with two

two will be canceled

now we have X plus 5 over 2 equals y

cubed

next let us eliminate the power 3 or

cube let us cube root both sides now the

power 3 and the cube root on the right

side will be canceled

cube root of x plus 5 over 2 equals y

or

the inverse function is the cube root of

x plus 5 over 2.

therefore the inverse of the function to

X cubed minus 5 is the cube root of x

plus 5 over 2.

that's it for this video I hope that you

learned how to solve the inverse of one

to one function thank you for watching

please like And subscribe