Warmup Alg 31 Jan 2012

Agenda• Don't forget about resources on

mrwaddell.net• Section 6.4: Inverses of functions

• Using “Composition” to prove inverse• Find the inverse of a function or relation

Practice from last class period’s assignment.

Section 6.4: Inverses of Functions

Vocabulary

Domain

Range

Inverse

Composition

The x values of the points

The y values of the points

A Function “flipped”

Putting one function inside another

Composition of Functions

If f(x) = 5x2 – 2x and g(x) = 4x

Then f(g(x)) is:

f(g(x)) = 5( )2 – 2( ) g(x) g(x)

f(g(x)) = 5( )2 – 2( ) 4x 4x

f(g(x)) = 5(16x2 ) – 2( 4x )

f(g(x)) = 80x2 – 8x

Composition of Functions 2

If f(x) = 5x2 – 2x and g(x) = 4x

Then g(f(x)) is:

g(f(x)) = 4( ) f(x)

g(f(x)) = 4( ) 5x2 – 2x

g(f(x)) = 20x2 – 8x

Composition of functions 3

If f(x)=2x and g(x) = 2x2+2 and h(x)= -4x + 3

Find g(h(2))

g(h(2)) = 2( )2 + 2 h(2)

g(h(2)) = 2( )2 + 2 -4(2) + 3

g(h(2)) = 2( )2 + 2 -8 + 3

g(h(2)) = 2( -5 )2 + 2 = 52

Composition of functions 3

If f(x)=2x and g(x) = 2x2+2 and h(x)= -4x + 3

Find h(g(2))

h(g(2)) = -4( ) + 3 g(2)

h(g(2)) = -4( ) + 3 2(2)2 +2

h(g(2)) = -4( ) + 3 2(4)+2

h(g(2)) = -4( 10 )+ 3 = -37

What we are doing

The inverse “flips” the picture over!

Inverse # 2

Find an equation for the inverse of:y = 2x + 3

First, switch the x and ySecond, solve for y.

x = 2y + 3-2y -2y

-2y +x = + 3 -x -x

-2y = -x + 3-2 -2 -2

y = ½ x – 3/2 That’s all there is to it.

Inverse #2

Now you try. Find the inverse of:

y = - ½x + 3

The inverse is: y = -2x + 6

Prove it! Here’s how.

Verifying an inverse is true.f(x) = - ½x + 3 and the inverse is: g(x) = -2x + 6

f(g(x)) = -½( ) + 3

f(g(x)) = -½( -2x + 6 ) + 3

f(g(x)) = x - 3 + 3

f(g(x)) = x

g(f(x)) = -2( ) + 6

g(f(x)) = -2(-½ x + 3) + 6

g(f(x)) = x - 6 + 6

g(f(x)) = x

Do (f◦g) and (g◦ f) and if they both equal “x” then they are inverses!

Non-linear inverse functions

The dashed line is the equation:

y = x

Notice the symmetry in the red and blue graphs!

Non-linear inversesThe dashed line is the equation:

y = x

Notice the symmetry in the red and blue graphs!

Checking Inverses #2

Can you show that

y = 2x + 3 and

y = ½ x – 3/2 are inverses of each other?

Do f(g(x)) and g(f(x)) and if they both equal “x” then they are inverses!

Hint: Call the first one “f(x)” and the second one “g(x)” and lose the “y’s”

Assignment

Section 6.4: 6-11,

15-20,

42-43

Do All, and pick 1 from each group to write complete explanation.