Inverse FUNctions MATH 3

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The range of the inverse of f(x) = x³

( -∞ , ∞ )

What is the inverse of the coordinate (2, -7)?

(-7, 2)

Identify the inverse: (3, 1) (2, 1) (4, 2) (5, 6)

(1, 3) (1, 2) (2, 4) (6, 5)

What is the inverse of the coordinate (1, 10)?

(10, 1)

Identify the inverse: (1, 3) (1, 2) (2, 4) (6, 5)

(3, 1) (2, 1) (4, 2) (5, 6)

Steps to solve algebraically for the inverse of the function

1. Replace f(x) with y in the equation for f(x) 2. Interchange x and y 3. Solve for y. 4. Replace y with f⁻¹(x)

What is the slope of the inverse equation of y = 2x + 4?

1/2

What is the slope of the inverse equation of y = 2/3 x - 6?

3/2

One-to-one Function

A property of functions where the same value for y is never paired with two different values of x (the function passes the horizontal line test)

Horizontal Line Test

A way to establish if a function is one-to-one when looking at a function's graph.

Vertical Line Test

A way to establish that a relation is a function.

Reflection about the line y = x.

A way to graphically see if two functions are inverses of each other.

Inverse Function

If a function is named f, this can be written as f⁻¹

Do the coordinates (-2, 5) and (-5, 2) represent inverse coordinates?

No

Is the inverse of the points below a function? (Hint: Find inverse first.) (3, 1) (2, 1) (4, 2) (5, 6)

No, because x=1 has two different outputs.

Domain Restriction

Omitting specific values from a relation's set of input values, commonly to ensure that a function's inverse is also a function.

Domain

The set of all input values of a relation.

Range

The set of all output values of a relation.

Do the coordinates (-3, 8) and (8, -3) represent inverse coordinates?

Yes

Is the inverse of the points below a function? (Hint: Find inverse first.) (1, 3) (1, 2) (2, 4) (6, 5)

Yes, because each output has a unique input.

The domain of the inverse of f(x) = x²

[ 0 , ∞ )

f⁻¹(x)= -3+³√(x+4)

f(x)= (x+3)³-4

f⁻¹(x)= -½(x)

f(x)= -2x

f⁻¹(x)= -4x

f(x)= -¼(x)

f⁻¹(x)= ½(x)

f(x)= 2x

f⁻¹(x)= ½(x) - 2

f(x)= 2x + 4

f⁻¹(x)= (x-4)³+1

f(x)= 4+³√(x-1)

f⁻¹(x)= x-25

f(x)= x+25

f⁻¹(x)= x-35

f(x)= x+35

f⁻¹(x)= x+14

f(x)= x-14

f⁻¹(x)= x+25

f(x)= x-25

f⁻¹(x)= x+6

f(x)= x-6

f⁻¹(x)+ (x+4)³-3

f(x)= ³√(x+3) -4

f⁻¹(x)= 4x

f(x)= ¼(x)

f⁻¹(x)= 2x - 10

f(x)= ½(x) + 5

f⁻¹(x)= 2x + 10

f(x)= ½(x) - 5

f(x)= 1/2x

f(x)=2x

f⁻¹(x)= ½(x) + 2

f(x)=2x-4

f(x) = 2x-3

f^(-1) (x) = (x+3)/2

f(x) = 6x + 5

f^(-1) (x) = (x-5)/6

f(x) = 1/2x - 1

f^(-1) (x) = 2x + 2

f(x) = x - 10

f^(-1) (x) = x + 10

f(x) = 4 - 2x

f^(-1) (x) =(x-4)/-2

f(x) = 5 - x

f^(-1) (x) =-x + 5

f(x) = (2x-6)/4

f^(-1) (x) =2x + 3

f(x) = 1/3 x+1

f^(-1) (x) =3x - 3

The inverse of f(x) = x + 11

f⁻¹(x) = x - 11

The inverse of f(x) = 2(x - 16)

f⁻¹(x) = ½x + 16

The inverse of f(x) = 2x - 16

f⁻¹(x) = ½x + 8

The inverse of f(x) = x^2 + 1

f⁻¹(x) = √(x-1)

The inverse of f(x) = (x + 1)^2

f⁻¹(x) = √x - 1

f(x)= -2x

f⁻¹(x)= -½(x)

f(x)= ½(x) + 5

f⁻¹(x)= 2x - 10

f(x)= ¼(x)

f⁻¹(x)= 4x

f(x)=2x-4

f⁻¹(x)= ½(x) + 2

f(x)=√(x+2) - 3

f⁻¹(x)=(x+3)²-2

f(x)=³√(x-3) + 5

f⁻¹(x)=(x-5)³+3

f(x)=x^2-7

f⁻¹(x)=√(x+7)

f(x)=(x-1)²+5

f⁻¹(x)=√(x-5)+1

What is the inverse of x squared?

square root of x

What does the range of a FUNction become in its inverse?

the domain

What does the domain become in the inverse of its FUNction?

the range

What is the inverse of the equation y = x/6 ?

y = 6x

What is the line of reflection for all inverse graphs?

y = x

What is the inverse of the equation y = x + 4 ?

y = x - 4


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