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domain: all possible x values
range all possible y values

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A function and its inverse is only a one-to-one function if:
- f(g(x))= x (for every x in the domain for g)
- g(f(x))= x (for every x in the domain for f)

For example:
f(x)= x³ +1
g(x) =∛(x-1)

To start, you would solve for the composition of the function.

So, if we were to solve f(x) first, it would become f(g(x))= (∛(x-1))³ +1

The cube root would be canceled by the power of three, so you would be left with x-1+1.  -1+1 would cancel out, so you would just be left with x.

To continue, you need to do the same thing to the other function g(x).
g(f(x)) = ∛(x³+1-1)

Both the cube root and the exponent of 3 would be canceled out as well as the +1-1. So, you would just be left with x.

f(g(x))= x
g(f(x))=x

Both functions are equal to x. Congrats! They are a one-to-one function!

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How to verify if two functions are inverses of each other

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graphically

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Use the Verticle Line test to identify if it is a function or not. If there is only 1 x value at each point on the verticle line then it's a function.
For each x value there is exactly 1 y value.
Two x-values can share a y-value
Two y-value's CANNOT share an x-value

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You can verify that two functions are inverses of each other by testing that for f(g(x))=x for every value of x in the domain of g, as well as for g(f(x))=x for every value of x in the domain of .

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Identifying Parent Functions and their Properties

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Identifying Parent Functions and their Properties

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combining functions through multiplication and division  

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combining functions through multiplication and division  

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multiplication: multiply like terms together, like expanding
division: set the function is a fraction and simplify

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Finding the inverses of functions algebraically

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Finding the inverses of functions algebraically

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1.) switch x and y
2.) solve for y
3.) factor out y if needed

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They are the maximums and minimums of the graph

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To find f(g(x)):
f(x)= 2x+3
g(x)=x+7
Plug in g(x) wherever there is an x in the f(x) equation
2(x+7)+3
"Solve" equation
2x+14+3
2x+17
So, f(g(x))=2x+17

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Combining Functions through Addition and Subtraction.

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Combining Functions through Addition and Subtraction.

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You have to add and subtract through the like terms of the functions given.

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Algebraically Even: For all x in the domain of f, f(-x)=f(x); To identify: plug in a # and its opposite and compare. Ex: f(x) = x^2
f(3) = 3^2=9
f(-3)=(-3)^2 = 9
*same=even*
Algebraically Odd: For all x in the domain of f, f(-x)=-f(x); to identify plug in a # and its opposite.Ex: f(x)=x^3
f(3)=(3)^3 = 27
f(-3)=(-3)^3 = -27 *opposites mean odd*

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Shifts are always either addition or subtraction, Stretches and shrinks are multiplication. H. are with the x and V. are outside. Reflections are represented with negatives.

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Master How to determine the extrema from a graph

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identifying if the inverse of a relation is a function

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identifying if the inverse of a relation is a function

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Decomposing Functions - Composition of Functions

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2. Identifying if the inverse of a relation is a function. 

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2. Identifying if the inverse of a relation is a function. 

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The inverse of a relation is only a function if it passes the horizontal line test.
If a relation passes the horizontal and vertical line test it is a "one-to-one" function.

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Finding the inverse of a function algebraically

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Finding the inverse of a function algebraically

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What is the domain of a function? | Functions | Algebra I | Khan Academy

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Identifying functions as even, odd, or neither

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Identifying functions as even, odd, or neither

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Even, Odd, or Neither Functions The Easy Way! - Graphs & Algebraically, Properties & Symmetry

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13. Transformation of Functions: How to tell the transformations of a function
https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:transformations/x2ec2f6f830c9fb89:trans-all-together/v/shifting-and-reflecting-functions

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Finding Local Maximum and Minimum Values of a Function - Relative Extrema

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How to find the new points and graphs with a chart (you can select the image and open it in a new window to view the image larger)

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Identifying if a relation is a function or not

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Identifying if a relation is a function or not

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Topics include:

1. Identifying if a relation is a function or not.

2. Identifying if the inverse of a relation is a function.

3. Finding domain and range of functions.

4. Identifying functions as being even, odd, or neither.

5. Identifying local extrema.

6. Stating intervals of increase and/or decrease.

7. Identifying parent functions and their properties (domain, range, even/odd/neither)

8. Finding the inverse of a function algebraically.

9. Verifying two functions are inverses.

10. Combing functions with addition, subtraction, multiplication, and division.

11. Composition of functions.

12. Decomposition of functions.

13. Transformations of functions.