RELATIONS AND FUNCTIONS

What is the inverse of g(x) = 1/x-1 and is it a function?

1/x + 1 , yes

Given: F(x) = 2x - 1; G(x) = 3x + 2; H(x) = x 2 Find F{G[H(2)]}.

27

Complete this activity. Given: f(x) = x + 2 and h(x) = 1/x-1 2f(x) - h(x) =

2x + 4 - 1/x-1

Given: F = {(0, 1), (2, 4), (4, 6), (6, 8)} and G = {(2, 5), (4, 7), (5, 8), (6, 9), (7, 5)} (F + G) (2) =

9

Answer the following question. Click on symbol to choose correct answer. Given: R = {(x, y): y = -x 2} What is the range of R?

y is lesser than or equal to 0

If G(x) = 5x- 2, find G-1(x).

(x + 2)/5

h(x) = 1/x+2 and k(x) = 3x - 4 h[k(x)] =

1/(3x - 2)

F(x)=x

Identity

Choose the ordered pairs that belong to the given relation. {(x, y): x 2 |y|3 - 5}:

(√3, -4) (-√3, 4)

Given: F(x) = x + 2 and G(x) = 3x + 5 (F G) (x) =

3x^2 + 11x + 10

Given: F(x) = 2x - 1; G(x) = 3x + 2; H(x) = x 2 Find F[G(x)] - F(x).

4x + 4

F(x)=0

Zero

Given: F(x) = 2x - 1; G(x) = 3x + 2; H(x) = x 2 Find H(x + a) - H(x).

a² + 2ax

I( x) = x; the function whose graph is the straight line through the origin with slope of 1

identity function

Given the relation of this table: n 1 0 2 s 1 0 3 What is the rule for this relation?

n(n+1)/2

Given: r = {√3, √5, √7, √13}. Why is r not a relation?

r is not a set of ordered pairs

Given: F(x) = 3x2+ 1, G(x) = 2x - 3, H(x) = x F(G(x)) =

12 x^2 - 36 x + 28

Given: F(x) = 3x 2+ 1, G(x) = 2x - 3, H(x) = x F(-2) =

13

Given: f(x) = x 2, g(x) = x + 6, h(x) = 7 Find f{g[h(x)] }.

169

Given: F(x) = 2x + 3, find F(x + h) - F(x)/h and simplify.

2

Given: F = {(0, 1), (2, 4), (4, 6), (6, 8)} and G = {(2, 5), (4, 7), (5, 8), (6, 9), (7, 5)} (F G) (2) =

20

f(x) = 3x + 2 and g(x) = x + 5 g[f(x)] =

3x + 7

Given: f = {(0, 1), (2, 4), (4, 6), (6, 8)} and g = {(2, 5), (4, 7), (5, 8), (6, 9), (7, 5)} Complete the ordered pair below for the function shown. f(x)/g(x)

6/7

If f(x) = 2x 2 - 3x + 1, find f(3) - f(2).

7

Perform the required operations on the following functions. Given: f(x) = 3 - x; g(x) = -2x Find f[g(2)].

7

Verify if f(x)=1/2x−2 and g(x)=2x+4 are inverses.

We must check that (f • g)(x) = x and (g • f)(x) = x. If both of these statements are true then f and g are inverses. (f o g)(x) = 1/2(2x+4)-2 =x+2-2 =x and (g o f)(x) = 2(1/2x-2)+4 =x-4+4 =x Since (f • g)(x) = x and (g • f)(x) = x, f and g are inverses.

G = {(0, 1)} Is G-1 a function and why?

Yes, each element in the domain has only one range value.

f(x) = {(1, 4), (2, 3), (5, 8), (4, 7)} Is f-1(x) a function and why?

Yes, each element in the domain has only one range value.

Use composition of functions to determine whether f(x) and g(x) are inverses of each other. Show all work for full credit. f(x) = 4/5x + 1 g(x) = 5x-5/4

f and , then f(x) and g(x) are inverses of each other. (f o g)(x) = 4/5(5x-5/4) + 1 = 20x-20/20 + 1 = 20x/20 - 20/20 + 1 = x-1+1 =x and (g o f)(x) = 5(4/5x + 1)-5/4 = 20/5x+5-5/4 = 4x+5-5/4 = 4x/4 =x Since and , f(x) and g(x) are inverses of each other.

Equal functions are two functions that have _____ domains, and, for each value in the domain set, _____ range values respectively.

same, equal

The inverse of a function occurs when _____.

the range and the domain are interchanged

Given: F(x) = 2x and G(x) = x2+ 2 Find (F + G)(x).

x 2 + 2 x + 2

Given: F(x) = 2x and G(x) = x2+ 2 Find (G - F)(x).

x 2-2 x + 2

Given: F(x) = 3x2+ 1, G(x) = 2x - 3, H(x) = x F(3) + G(4) - 2H(5) =

23

Given: F(x) = 2x - 1; G(x) = 3x + 2; H(x) = x 2 Find H(x+a)-H(x)/a

2x + a

Given: F(x) = x 2, find F(x + h) - F(x)/h and simplify.

2x + h

Perform the required operations on the following functions. Given: f(x) = 3 - x; g(x) = -2x Find g[f(x)].

2x - 6

f(x) = x 2+ 6 and g(x) = 2x - 1 g[f(x)] =

2x² + 11

Given: F(x) = 3x 2+ 1, G(x) = 2x - 3, H(x) = x F(x) + G(x) =

3 x^2 + 2 x - 2

Given: f(x) = x² and g(x) = 2x + 1, find f[g(x)].

4x² + 4x + 1

Given: F(x) = 2x - 1; G(x) = 3x + 2; H(x) = x 2 Find G[H(1)].

5

Given: f(x) = x² and g(x) = x + 1, find g[f(-2)].

5

Given: f(x) = x 2, g(x) = x + 6, h(x) = 7 Find g{f[h(x)] }.

55

Given: f(x) = x 2, g(x) = x + 6, h(x) = 7 Find h{g[f(x)] }.

7

K = {(2, 1), (3, 2), (4, 1), (5, 3)} Is K-1 a function and why?

No, each element in the domain does not have one range value

G = {(5, 3), (2, 3), (6, 4)} Is G-1 a function and why?

No, each element in the domain does not have one range value.

Given: B = {(4, 2), (4, -2), (16, 4), . . .} Is B a function and why?

No, two ordered pairs in this list have a repeat of the domain element.

Given: A = {(2, 3), (5, 1), (-3, -2), (0, 3)} A is a subset of ______.

R X R

Prove that and g(x) = 2x - 3 are inverses.

We must check that (f • g)(x) = x and (g • f)(x) = x. If both of these statements are true then f and g are inverses. (f o g)(x) = (2x - 3)+3/2 = 2x-3+3/2 =2x/2 =x and (g o f)(x) = 2(x+3/2)-3 =x+3-3 =x Since (f • g)(x) = x and (g • f)(x) = x, f and g are inverses.

The domain of {(x, y): y = 2x 2 + 1 is

all real numbers

Given: F(x) = 3x2+ 1, G(x) = 2x - 3, H(x) = x H-1(x) =

x

Given: f(x) = x + 2 and g(x) = 3x + 5 f(x)/g(x)=

x+2/3x+5

What is the inverse of f(x) = 3x + 6 and is it a function?

x-6/3 , yes

D = {[x, f(x)]: (-1, 3), (0, 2), (1, 3), (2, 6), (3, 11)} Write the rule for f(x).

x^2 + 2

Given: f(x) = x + 2 [f(x)]^2 =

x^2 + 4x +4

Given: F(x) = 2x - 1; G(x) = 3x + 2; H(x) = x 2 Find F(x) + G(x) + H(x).

x² + 5x + 1

g(x) = x + 7 and h(x) = x 2+ 1 g[h(x)] =

x² + 8

Given: A = {(0, 0), (2, 1), (3, 1.5), (4, 2), . . . } Write a rule for A.

y = x/2

Given: B = {(4, 2), (4, -2), (16, 4), . . . } Write a rule for B.

y² = x

Z( x) = 0; the constant function whose graph is the line Y = 0, the x-axis

zero function

What is the domain of the relation {(x,y):y = x(x-3)/(x+4)(x-7)}?

{ x: x E R, x (cancel) = -4, x (cancel) = 7}

Given: A = {(2, 3), (5, 1), (-3, -2), (0, 3)} What is the range of A?

{-2, 1, 3}

Given: A = {(2, 3), (5, 1), (-3, -2), (0, 3)} What is the domain of A?

{-3, 0, 2, 5}

Given: F = {(0, 1), (2, 4), (4, 6), (6, 8)} and G = {(2, 5), (4, 7), (5, 8), (6, 9), (7, 5)} Find the common domain of F and G.

{2, 4, 6}

{(x, y): y = √x-3 The domain of the set above is represented by _____.

{x: x ≥ 3}

g(x) = 2x2 and h(x) = √x^2 + 1.What is (g o h)^-1 and is it a function?

±√x-2/2 , No

What is the inverse of f(x) = x2 + 5 and is it a function?

±√x-5 , No

Given: F(x) = 5x - 6 and G(x) = x - 4 (FG)-1 =

( x + 26)/5

Answer the following question. Click on symbol to choose correct answer. Given: R = {(x, y): y = -x 2} What is the domain of R?

x E R

Verify if f(x) = 5-3x/2 and g(x) = 5-2x/3 are inverses.

We must check that (f • g)(x) = x and (g • f)(x) = x. If both of these statements are true then f and g are inverses. (f o g)(x) = 5-3(5-2x/3)/2 = 5 - (15/2) + (6x/3)/2 =5-5+2x/2 =2x/2 =x and (g o f)(x) = 5-2(5-3x/2)/3 = 5-(10/2)+(6x/2)/3 =5-5+3x/3 =3x/3 =x Since (f • g)(x) = x and (g • f)(x) = x, f and g are inverses.

Given: A = {(0, 0), (2, 1), (3, 1.5), (4, 2), . . .} Is A a function and why?

Yes, no two ordered pairs in this list has a repeat of the domain element.

Given: F(x) = 5x - 6 and G(x) = x - 4 G -1F -1 =

( x + 26)/5

Given: F(x) = 3x2+ 1, G(x) = 2x - 3, H(x) = x G-1(x) =

( x + 3)/2

Given: F(x) = x + 2 and G(x) = 3x + 5 (F + G) (x) =

4x + 7

F(x)=n

Constant

(F o G)(x)

F[G(x)]

What is the inverse of f(x)? Show all work for full credit. f(x) = 2/x-6

Let f(x) = 2/x-6 , or y = 2/x-6 Then for f^-1(x), we have x = 2/y-6 x = 2/y-6 x(y-6) = 2 y-6 = 2/x y - 2/x + 6 Thus, f^-1 (x) = 2/x + 6

Given: f(x) = x + 2 and h(x) = 1/x-1 Evaluate the following function at x=2. [h (x)]^2/f(x)

1/4

Complete the set of ordered pairs for the relation. {(x, y): y = 2 |x + 1| and x {-2, -1, 0, 1, 2}}.(-2, __)

2

Complete the missing portion of the equation. f(x) = 1/x^2 and g(x) = 1/2x +4 f[g(x)] = 4x 2 _____

+ 16x + 16

Given: F = {(0, 1), (2, 4), (4, 6), (6, 8)} and G = {(2, 5), (4, 7), (5, 8), (6, 9), (7, 5)} (F - G) (6) =

-1

Given: F(x) = 2x and G(x) = x2+ 2 Find F/G(-1).

-2/3

Given: F(x) = x + 2 and G(x) = 3x + 5 (F - G) (x) =

-2x - 3

Given: f(x) = 3 - x; g(x) = -2x Find g[f(-1)].

-8

Using range values of some given function as domain values of some other given function to find new range values

composition of functions

C( x) = N; a function whose graph is a horizontal line Y = N

constant function

Complete this activity. Include all of your work in your final answer. Submit your solution. Given: f(x) = x2 + 2x + 1, find f(x + h) and simplify.

f(x) = x2 + 2x + 1 f(x + h) = (x + h)2 + 2(x + h) + 1 = x2 + 2xh + h2 + 2x + 2h + 1