Algebra 3 Assignment Sheet Functions, Fog, Gof,

(1) Assignment # 1 – Functions, Domains

(2) Assignment # 2 – Composition of Functions

(3) Assignment # 3 – Inverse Functions

(4) Review

(5) Test

Functions

I Vocabulary

a) Relation: a set of ordered pairs where the first number is the domain and the second is the range.

b) Function: The set of (x, y) pairs such that for every value of x there is a unique value of y A 1 -1 correspondence between 2 sets such that for each value in the domain set there is only 1 value in the range set.

b) domain: “_x__” values c) range: “_y_” values

II Functions ? Y/N Domain

A. a) (2,4) (8,7) (8,8) _____ ______________

b) (1,-2) (3,4) (5,-2) _____ ______________

c) 2y x 5x 6 _____ ______________

d) y = |x| _____ ______________

e) 2y = 8 _____ ______________

f) y x _____ ______________

g) 2

y = - x + 2 _____ ______________

h) 2

2xf(x)

x 4 _____ ______________

f(x) is the same as y

III Restrictions

1) Denominator 0 5 + x

x,

2 -3x

x -5,

2

x

x + 5x - 6

2) negative number

Find the domain of the following please.

a) f(x) x 2 b) f(x) 2x 5 c) x

f(x) x 2

d) 2y = x x 12 e) 2

2

2 63

8 15

x xy =

x x

IV Evaluating functions for y when given a value of x

Evaluating: Substituting in numbers y = f(x)

2

2

2

2

2 3

(2) 2 2 2 3 11

1 1 2 1 3 2

( ) 2 3

y x x

f

f

f x x x 2( ) 2 3f x x x

1)

f(-1) = f(0) =

even roots

x x x

2f (x) x 2x 3

2)

Find f (x + 2)

3) If f(x) = 2 3 1x x g(x) = x - 5

a) find f(x + 2) b) find 3f( x) g(2x)

c) f(x) g(x) d) f(g(x))

( called a composition of functions )

2f(x) x 2x 3

Algebra 3 Assignment # 1

(1) 4 3 2

f x = 3x 16x 7x 28x 13 , 2

g x = x + 2x 4 . Find each of the following.

(a) g 4 (c) f g 3

(b) f 6 (d)

(2) Determine whether each of the following defines y as a function of x. If it is a function, find the

domain please.

(a) y 5x 4 (f) y 5 4x

(b) 2

y x + 2x 3 (g) 2

y x 5x 6

(c) 2 2

x + y = 1 (h) 2

xy

x + 1

(d) 3x + 5

y 2x 3

(i) 5 2x

y 3x + 4

(e)

3

3

x + 8y =

x 7x + 6 (j)

3 2y x 9x + 23x 15

(3) Let the function f be defined by f x = x + 1. If 2 f x = 20 , find f 3 x .

(4) Let the function f be defined by f x = x + c , where c is constant. If f 2 = 10 , find the value

of the constant c.

Answers

(1) (a) 20 (c) 27

(b) –1 (d) –3 – 12i

(2) (a) Real (f) 54

x

(b) Not a function (g) x 1 or x 6

(c) Not a function (h) Real

(d) 32

x (i) 543 2

x

(e) x 1 , 2 , 3 (j) 1 x 3 or x 5

(3) 28 (4) 8

The remainder when f x is

divided by x i

Composition of functions

(A) Let 22 2f(x) x 1 and g(x) = x

Find f(2) Find f(g(x))

Find g( 2)

Find f( g(2))

Find g(f(2))

(called a composition)

(B) If 22 1

1f(x) x , g(x)=x and h(x)=

3x

Find f g h(x)

If 6

4

x x+1f(x) and g(x)=

x x+5 find f g

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

DOMAIN:

Find f g x and g f x for each of the following:

e) f x 4x 3

g x x 6 f)

2

f x x 3

g x x 2x 4

If 2

1f (x) x and g(x) =

x 2, then find the domain of g f

Algebra 3 Assignment # 2

Composition of Functions

(1) Find each of the following numbers, given the functions below.

1 2x xf ; 2 x 2x xg 2 ; 1 x xh 2

(a) 2hf (b) 1hg

(c) 25fg (d) 3gfh

(2) Find xgf and xfg for each of the following please.

(a) 5 4x xg

1 2x 3x xf 2

(b)

1 3x

2 x xg

3 4x

5 2x xf

(3) Find xg if 5 2x

1 3x xf , and

11 12x

9 x xgf .

(4) Find xff if 2 3x

3 2x xf

(5) Find all values of x such that xfg xgf if

2. 3x xg and , 2 3x 2x xf 2

Algebra 3 Assignment # 2

Answers

(1) (a) 3 (b) 4

(c) 89 (d) 26

(2) (a) 64 112x 48x xgf 2 (b)

11 x5

1 17x xgf

9 8x 12x xfg 2

18 2x

1 10x xfg

(3) 3 2x

2 x xg

(4) xff = x

(5) x = 1

Intro to Inverse

1. Make a table of points for the function. 2. Make a table of points for the function

y 2x 1 x 1

y2

3. Do you notice anything interesting about the two tables?

4. Draw both functions, in different colors, on the axis here.

5. Add the function y = x to the graph, drawing

it as a dotted line. (Like an asymptote)

6. Make a table of points for the function. 7. Make a table of points for the function

2y x 2 0x y x 2

8. Do you notice anything interesting about the two tables?

9. Draw both functions, in different colors, on the axis here.

10. Add the function y = x to the graph, drawing

it as a dotted line. (Like an asymptote)

11. Make a table of points for the function. 12. Make a table of points for the function xy 2 yx 2

13. Do you notice anything interesting about the two tables?

14. Draw both functions, in different colors, on the axis here.

15. Add the function y = x to the graph, drawing

it as a dotted line. (Like an asymptote)

Go back and look at all three drawings.

What does the line y = x look like it does in each of these pictures? Describe in 2 to 3 sentences what you

see.

4.5 Inverse functions ( 1f and f ) 1f is not 1

f

NOTES:

All functions have inverses, but the inverse is not necessarily a function.

If an inverse of a function f is also a function we denote it by 1f . ( Read as f inverse)

To obtain the inverse of a relation, you interchange the coordinates of each ordered pair.

(Switch the domain and range)

To obtain the inverse of a function, the function must be a 1 – 1 function, and is found the same way.

Therefore, the domain of the function is the range of the inverse.

A. Finding inverses

Switch the domain “x” and the range “y”

Then solve for “y”

EX. f(x) = 2x - 5 y = 2x – 5

switch x = 2y – 5

+5 + 5

solve

x 5 2y

x 5y

2

this is the inverse

B. If f(x) and g(x) are inverses of each other

then f[ g(x)] = x and g[f(x)] = x

Ex. If f(x) =2x - 3 g(x) = x 3

2

x 3 x 3

f 2 32 2

= x + 3 – 3 = x

(f(x))

2x 3 3 2x

g 2x 32 2

x

g(x)

C Practice Find 1f (x) for each of the following:

a) (2,3) (4,-2) (0,2)

b) 3h x x 1

c) 2

f x x 2 3

d) 5x 1

g x2

e) f x 3x 2 (uh oh, aren’t there restrictions?)

Algebra 3 Assignment # 3

Inverse Functions

(1) Find xf 1 for each of the following please.

(a) 3 5x xf (e) 7 5x xf

(b) x

4 xf (f) 2 5 4x xf

(c) 2 5x

2 3x xf (g)

3 5 4x xf

(d) 7 2x

2 7x xf (h) 2 8 5x xf 3

(2) 3 2x

5 6x xf , x xfg . Find xg .

(3) 2 x xf . Find xf 1, and sketch a graph of xf and xf 1

on the same set of axes.

Algebra 3 Assignment # 3

Answers

(1) (a) xf 1 =

5

3 x (e) xf 1

= 5

7 x 2

(b) xf 1 =

x

4 (f) xf 1

= 4

1 4x x 2

(c) xf 1 =

3 5x

2 x2 (g) xf 1

= 4

5 x3

(d) xf 1 =

7 2x

2 x7 (h) xf 1

= 5

8 2 x 3

(2) xg = 6 2x

5 x3

Algebra 3 Assignment # 4 Review Worksheet

(1) Find each of the following numbers, given the functions below.

2x x xf 2 ; 3x xg ; 1 x xh

(a) 3hf (b) 0hg

(c) 8ghf (d) 8hfg

(2) Find the domain of each of the following functions please.

(a) 2

f x = 24x 29x 4 (b) 2

x 1 f x =

x 9

(c) 3 2

5x + 2f x =

x 4x + x + 6 (d)

2 x + 3 x 1

f x = x 5

(3) Find xgf and xfg for each of the following.

(a) 3 x xg

1 2x xf

2 (b)

1 2x

1 x xg

2 3x

3 2x xf

(4) Find xf 1 for each of the following.

(a) 7 5x xf (b) 4 5 x2 xf 3

(c) 5 2x

2 3x xf (d) 1 x5 xf

(5) Find all values of x for which xfg xgf if 5 x xf and

3 4x 2x xg 2.

(6) Find xg , if 2 x

1 2x xf and

1 3x

1 6x xgf .

(7) Let f x = 5x 7 . Find all values of such that 3f = 10.

2

Algebra 3 Assignment # 4 Review Worksheet Answers

(1) (a) 0 (b) 3

(c) 15 (d) 9

(2) (a) 1 48 3

x or x (b) x 1 and x 3

(c) x 1 , 2 , 3 (d) x 3 or x = 1 or x > 5

(3) (a) 5 2x xgf 2 , 2 4x 4x xfg 2

(b) 5 x

1 8x xgf ,

8 x

1 5x xfg

(4) (a) 5

7 x xf 1

(b) 2

5 4 x xf

31

(c) 3 2x

2 5x xf 1

(d) 5

1 x xf

21

(5) 4

15

(6) 1 3x xg

(7) 3130

Algebra 3/Trig

Extra Review – 4-4, 4-5, 5-1 to 5-3

A.

Find f g and g f if

1. f(x) 3x 1 , 2

1g(x)

x 1 2.

3f(x) 1 x , 3g(x) 1 x

3. If the function 2x 1

f(x)x 2

and 6x 1

f(g(x))3x 1

, find g(x).

Find 1f (x) for the following.

4. 3f(x) 2x 5 4 5. 2x 1

f(x)x 2

6. f(x) 5x 1

ANSWERS

1. 2

2

2 xf g

x 1

2

1g f

9x 6x 2

2. f g g f x

3. g(x) 3x 1

4. 3 2

1 x 12x 48x 159f (x)

2 5. 1 1 2x

f (x)x 2

6. 2

1 x 1f (x)

5