5.1 Composite Functions5.1 Composite Functions

GoalsGoals1.1. Form f(g(Form f(g(xx)) = (f )) = (f g) (g) (xx))

2.2. Show that 2 Composites areShow that 2 Composites are EqualEqual

x

Evaluate a Composite FunctionEvaluate a Composite Function

(text p. 393)(text p. 393)

f (x) = 2xf (x) = 2x22 – 3 – 3

g(x) = 4xg(x) = 4x

Find fFind f00g (1) and gg (1) and g00g (-1)g (-1)

Find a Composite Function (text p. 393)Find a Composite Function (text p. 393)

Given,Given,

f(x) = xf(x) = x22+ 3x -1+ 3x -1

g(x) = 2x + 3g(x) = 2x + 3

Find fFind f g and gg and gff

What do you notice from the previous What do you notice from the previous example?example?

ff○○g (x) g (x) ≠ g≠ g○○f (x)f (x)

Show that 2 composites are equal Show that 2 composites are equal

(text p. 395)(text p. 395)

Given, functions f and g such that,Given, functions f and g such that,

f(x) = 3x f(x) = 3x 4 4 g(x) = g(x) = ⅓(x + 4)⅓(x + 4)

Show that Show that ff○○g (x) g (x) = g= g○○f (x) = xf (x) = x

for every x.for every x.

SOLUTIONSOLUTION

(f (f g) (g) (xx)) = f(g( = f(g(xx)) )) = f(= f(⅓( ⅓( x x + 4+ 4)))) = 3(= 3(⅓(⅓(xx + 4 + 4))) - 4) - 4 = = xx + 4 – 4 + 4 – 4 = = xx

(g(gf) (f) (xx) = g(f() = g(f(xx)))) = g(= g(3 3 x x + 4+ 4)) = = ⅓⅓[(3[(3xx - 4 - 4)+)+44]] = = ⅓ ⅓ (3x)(3x)

= = xx

When f(g(x)) = g (f (x))When f(g(x)) = g (f (x))

We say that the functions areWe say that the functions are

INVERSE OF EACH OTHERINVERSE OF EACH OTHER

5.2 Inverse Functions5.2 Inverse Functions

GoalsGoals1.1. Determine the Inverse of a FunctionDetermine the Inverse of a Function

2.2. Obtain the Graph of the Inverse Obtain the Graph of the Inverse from a given Graph from a given Graph

3.3. Find Find ff--1 1 the inverse functionthe inverse function

x

Example 1 (text p. 400)Example 1 (text p. 400)

Example 2 (text p. 401)Example 2 (text p. 401)

Discussion of example 5 text p. 405Discussion of example 5 text p. 405

Finding the Inverse Function text p. 407Finding the Inverse Function text p. 407

In groupsIn groups

1.1. # 51# 51 p. 411p. 411

2.2. # 23 # 23 p. 410p. 410

3.3. # 9 item c # 9 item c p. 397p. 397

4.4. # 47# 47 p. 398p. 398