Inverse of One-to-One Function | Grade 11- General Mathematics
Summary
TLDRThis 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
📘 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.
Mindmap
Keywords
💡Inverse Function
💡One-to-One Function
💡Interchange Variables
💡Transposing
💡Solving for y
💡Cross Multiply
💡Grouping Terms
💡Dividing by a Coefficient
💡Cube Root
💡Cubing
💡Rational Expression
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
5.0 / 5 (0 votes)