Inverse FUNctions MATH 3
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
