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7.4 Inverse Functions

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7.4 Inverse Functions

p. 422

- 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

- 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.

- Steps: -switch x & y
-solve for y

y = -3x+6

x = -3y+6

x-6 = -3y

- 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”

- 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.

- 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!

(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.

- 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.

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

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.

y=2x3

x=2y3

OR, if you fix the tent in the basement…

Inverse is a function!