Inverse Functions & Graphs

Inverse Functions & Graphs

KY.HS. A.2 Use the structure of an expression to identify ways to rewrite it and consistently look for opportunities to rewrite expressions in equivalent forms.

KY.HS. F.9. Find the inverse functions

a. Given the equation of an invertible function, find its inverse.

• In mathematics, an inverse is a function that serves to “undo” another function.

• Notation: f^-1(x)

• With inverses the composition = x

   $f\left(f^{-1}\left(x\right)\right)=x\$ 

•  $or\ f^{-1}\left(f\left(x\right)\right)=x$  

lnverse Fuctions

Domain of f is the range of f^-1

Range of f is the domain of f^-1

*Points (x, y) of f(x) becomes

(y, x) on f^-1(x)

x's and y's switch places

Inverse Graphs

• Symmetrical across the line y=x

• Points on f(x) like (2, 4)

• becomes on f^-1(x) as (4, 2)

Multiple Choice

Are these functions inverses?

• Yes
• No

Multiple Choice

Are the blue and red graphs inverse functions?

• yes
• no
• no way to tell

Multiple Choice

Are these inverse functions?

• no
• yes
• no way to tell

Multiple Choice

What is the inverse of the points
(1,3)(2,4)(6,8)

• (3,1)(4,2)(8,6)
• (-3,-1)(-4,-2)(-8,-6)
• (1,3)(2,4)(6,8)
• (-1,-3)(-2,-4)(-6,-8)

Multiple Choice

What is the RANGE of the INVERSE function?

• {-5, 1, 3, 4}

• {-4, -3, -1, 5}

• {-4, -2, 0, 7}

• {-7, 0, 2, 4}

Multiple Choice

If f^-1(-25) = 7 then what is f(7) = 

• 25
• 7
• No enough information. 
• -25

Multiple Choice

What is the inverse of the given coordinates: (1, 2) (3, 8) (-3, 6)

• (1, 2) (3, 8) (-3, 6)
• (1, 2) (8, 3) (-3, 6)
• (-1, -2) (-3, -8) (3, -6)
• (2, 1) (8, 3) (6, -3)

Multiple Choice

The inverse has been reflected over which line?

• y = x
• y =1
• y = 0
• y = x + 1

Finding an Inverse

• Switch x's and y's

• Solve for the y

• Substitute notation for the new "y=" equation to f^-1(x)=

Multiple Choice

Was the inverse function found correctly?

• no,

• yes

Multiple Select

 $g\left(x\right)=-7x^5+7$  Find the inverse of the function.

•  $g^{-1}\left(x\right)=^5\sqrt{\frac{x-7}{7}}$  

•  $g^{-1}\left(x\right)=-^5\sqrt{\frac{x-7}{7}}$  

•  $g^{-1}\left(x\right)=^5\sqrt{-\frac{x-7}{7}}$  

•  $g^{-1}\left(x\right)=-^5\sqrt{\frac{x-7}{7}}$  

Multiple Choice

When finding the inverse of a relation, 
1st you replace f(x) with y
2nd switch the x's and y's
What do you do next?

• solve for y
• replace y with f^-1(x)
• solve for x
• find the greatest common factor

Multiple Choice

What is the FIRST step I need to take to solve for x in the following equation?

• Add 9 to both sides

• Take the square root of both sides

• Divide both sides by 49

• Multiply both sides by 9

Multiple Choice

Find the inverse of
f(x) = 1/4x - 7

• f^-1(x) = 4x + 7
• f^-1(x) = -4x + 28
• f^-1(x) = -4x - 7
• f^-1(x) = 4x + 28

Multiple Choice

Find the inverse of f(x) = -4x - 12

• f^-1(x) = 4x - 3
• f^-1(x) = -1/4x - 3
• f^-1(x) = 1/4x + 3
• f^-1(x) = -4x - 3

Multiple Choice

FInd the inverse of the function.
 $f\left(x\right)=\frac{3x-15}{2}$  

•  $f^{-1}\left(x\right)=-x-5$  

•  $f^{-1}\left(x\right)=3x+5$  

•  $f^{-1}\left(x\right)=\frac{15+2x}{3}$  

•  $f^{-1}\left(x\right)=\frac{2}{3}x-\frac{19}{3}$