4-5 INVERSE FUNCTIONS
Objectives:
1. Find the inverse of a function, if it exists.
INVERSE EXAMPLE
Conversion

formulas come in pairs, for

example:

These

formulas “undo” each other, so
they are invers...
DEFINITION OF INVERSES
Two

functions f and g are inverses if:
f(g(x)) = x and g(f(x)) = x

To

check if two functions a...
EXAMPLE 1
If

and

show that f and g are inverses.
YOU TRY!
Show

that
are inverses.

and
INVERSE NOTATION
The

inverse of f is written f -1

f -1(x)

Note:

is the value of f -1 at x

is not
FINDING INVERSES
graph of f -1 is the reflection of f
over the line y = x
Can be found by
switching x and y
in the ordere...
EXAMPLE 2
f(x) = 4 – x2 for x 0.
Sketch the graph of f and f -1 (x)
Find a rule for f -1 (x)
Let
YOU TRY!
g(x) = (x – 4)2 – 1 for x 4.
Sketch the graph of g and g -1 (x)
Find a rule for g -1 (x)
Let
EXAMPLE 3
Suppose

a function f has an inverse.
If f(2) = 3, find:

f -1

(3)

f(f -1(3))
f -1(f(2))
YOU TRY!
Suppose

a function g has an inverse.
If g(5) = 1, find:

g -1

(1)

g -1(g(5))
g(g -1(1))
DO ALL FUNCTIONS HAVE
INVERSES?
the graph of y = x2 over the
line y = x.
Is the result a function?
Reflect
ONE-TO-ONE
Only

functions that are one-to-one
have inverses.
One-to-one means each x value has
exactly one y value and ...
EXAMPLE 4
Which

functions are one-to-one?
Which have inverses?
YOU TRY!
Is h(x) one-to-one?
 Does it have an inverse?

EXAMPLE 5
State

whether each function has an
inverse. If yes, find f -1 (x) and show
f(f -1(x)) = f -1(f(x)) = x




YOU TRY!
Does

have an
inverse? If so, find f -1 (x).
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4 5 inverse functions

  1. 1. 4-5 INVERSE FUNCTIONS Objectives: 1. Find the inverse of a function, if it exists.
  2. 2. INVERSE EXAMPLE Conversion formulas come in pairs, for example: These formulas “undo” each other, so they are inverses.
  3. 3. DEFINITION OF INVERSES Two functions f and g are inverses if: f(g(x)) = x and g(f(x)) = x To check if two functions are inverses, perform both compositions and make sure both equal x.
  4. 4. EXAMPLE 1 If and show that f and g are inverses.
  5. 5. YOU TRY! Show that are inverses. and
  6. 6. INVERSE NOTATION The inverse of f is written f -1 f -1(x) Note: is the value of f -1 at x is not
  7. 7. FINDING INVERSES graph of f -1 is the reflection of f over the line y = x Can be found by switching x and y in the ordered pairs. Find equation of f -1 by switching x and y in the equation and solving for y. The
  8. 8. EXAMPLE 2 f(x) = 4 – x2 for x 0. Sketch the graph of f and f -1 (x) Find a rule for f -1 (x) Let
  9. 9. YOU TRY! g(x) = (x – 4)2 – 1 for x 4. Sketch the graph of g and g -1 (x) Find a rule for g -1 (x) Let
  10. 10. EXAMPLE 3 Suppose a function f has an inverse. If f(2) = 3, find: f -1 (3) f(f -1(3)) f -1(f(2))
  11. 11. YOU TRY! Suppose a function g has an inverse. If g(5) = 1, find: g -1 (1) g -1(g(5)) g(g -1(1))
  12. 12. DO ALL FUNCTIONS HAVE INVERSES? the graph of y = x2 over the line y = x. Is the result a function? Reflect
  13. 13. ONE-TO-ONE Only functions that are one-to-one have inverses. One-to-one means each x value has exactly one y value and each y has exactly one x Can check using horizontal line test.
  14. 14. EXAMPLE 4 Which functions are one-to-one? Which have inverses?
  15. 15. YOU TRY! Is h(x) one-to-one?  Does it have an inverse? 
  16. 16. EXAMPLE 5 State whether each function has an inverse. If yes, find f -1 (x) and show f(f -1(x)) = f -1(f(x)) = x  
  17. 17. YOU TRY! Does have an inverse? If so, find f -1 (x).
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