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7.4 Inverse Functions. p. 422. Review from chapter 2. Relation – a mapping of input values (x-values) onto output values (y-values). Here are 3 ways to show the same relation. x y -2 4 -1 1 0 0 1 1. y = x 2. Equation Table of values Graph. x y -2

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review from chapter 2
Review from chapter 2
  • Relation – a mapping of input values (x-values) onto output values (y-values).
  • Here are 3 ways to show the same relation.

x y

-2 4

-1 1

0 0

1 1

y = x2

Equation

Table of values

Graph

slide3
x y
  • -2
  • -1
  • 0 0
  • 1 1

x = y2

  • Inverse relation – just think: switch the x & y-values.

** the inverse of an equation: switch the x & y and solve for y.

** the inverse of a table: switch the x & y.

** the inverse of a graph: the reflection of the original graph in the line y = x.

ex find an inverse of y 3x 6
Ex: Find an inverse of y = -3x+6.
  • Steps: -switch x & y

-solve for y

y = -3x+6

x = -3y+6

x-6 = -3y

inverse functions
Inverse Functions
  • Given 2 functions, f(x) & g(x), if f(g(x))=x AND g(f(x))=x, then f(x) & g(x) are inverses of each other.

Symbols: f -1(x) means “f inverse of x”

ex verify that f x 3x 6 and g x 1 3 x 2 are inverses
Ex: Verify that f(x)=-3x+6 and g(x)=-1/3x+2 are inverses.
  • Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses.

f(g(x))= -3(-1/3x+2)+6

= x-6+6

= x

g(f(x))= -1/3(-3x+6)+2

= x-2+2

= x

** Because f(g(x))=x and g(f(x))=x, they are inverses.

to find the inverse of a function
To find the inverse of a function:
  • Change the f(x) to a y.
  • Switch the x & y values.
  • Solve the new equation for y.

** Remember functions have to pass the vertical line test!

ex a find the inverse of f x x 5
Ex: (a)Find the inverse of f(x)=x5.

(b) Is f -1(x) a function?

(hint: look at the graph!

Does it pass the vertical line test?)

  • y = x5
  • x = y5

Yes , f -1(x) is a function.

horizontal line test
Horizontal Line Test
  • Used to determine whether a function’s inverse will be a function by seeing if the original function passes the horizontal line test.
  • If the original function passes the horizontal line test, then its inverse is a function.
  • If the original function does not pass the horizontal line test, then its inverse is not a function.
ex graph the function f x x 2 and determine whether its inverse is a function
Ex: Graph the function f(x)=x2 and determine whether its inverse is a function.

Graph does not pass the horizontal line test, therefore the inverse is not a function.

ex f x 2x 2 4 determine whether f 1 x is a function then find the inverse equation
Ex: f(x)=2x2-4 Determine whether f -1(x) is a function, then find the inverse equation.

y = 2x2-4

x = 2y2-4

x+4 = 2y2

OR, if you fix the tent in the basement…

f -1(x) is not a function.

ex g x 2x 3
Ex: g(x)=2x3

y=2x3

x=2y3

OR, if you fix the tent in the basement…

Inverse is a function!

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