Lesson 1: Functions as Models

A function happen to be a simple mathemical model or a piece of larger model.

Recall that a functon is just a rule or law f, that expresses the dependency of a

variable y, on another variable x.

Example 1: The graph of a function f is shown below:

(a) Find the values f(1) and f(5).

(b) What is the domain and range of f ? Solution:

(a) We see from the graph that the point (1,3) lies on the graph of f, so the

value of f at 1 is f(1) = 3. While x = 5, the graph lies about 0.7 unit below the

x- axis. Therefore, we estimate that f(5) = -0.7.

(b) Notice that f(x) is dened when 0 x 7, so the domain of f is the

closed interval 0,7. See that f takes on all values from the interval -2 to 4, so

the range of f is

-2y 4 = -2,4

Example 2: Sketch the graph and nd the domain and range of the func-

tions:

(a) f(x) = 2x- 1

(b) g(x) = x2

Solution:

(a)The equation of graph is y = 2x-1, and recognize this as being the equa-

tion of a line with slope 2 and y-intercept -1.

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Recall: The slope- intercept form of the equation of a line y = mx + b.

This enables us to sketch the graph below. The expression is dened for all

real numbers, so the domain of is the set of all real numbers, which we denote

by R. The graph shows that the range is also R. (b) Since g(2) = 2

2

= 4 and g(-1) = ( 1) 2

= 1, we could plot the points (2,4)

and (-1,1), together with a few other points on the graph, and join them to pro-

duce the graph below. The equation of the graph is y = x2

, which represents a

parabola. The domain of g is R. The range of g consists of all values of g(x),

and that is the all numbers of the form x2

. But x2

0 for all numbers x and

any positive number of y is a square. Therefore, the range of g is y| y 0 =

0, 1). This can also be seen from the gure below. Example 3:

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Lesson 2: Evaluating Functions

To evaluate a function

1.Substitute the given value in the function of x.

2.Replace all the variable xwith the value of the function.

3.Then compute and simplify the given function.

Example 1: Given the function: f(x ) = 2 x+ 1, nd f(6).

Substitute 6 in place holder x,

f(6) = 2 x+ 1

Replace all the variable of xwith 6,

f(6) = 2(6) + 1

Then compute function. f(6) = 12 + 1

f (6) = 13

Therefore, f(6) = 13. It can also write in ordered pair (6,13).

Example 2: Given the function f(x ) = x2

+ 2 x+ 4 when x= 4. Substitute

-4 in the place holder x,

f( 4) = x2

+ 2 x+ 4

Replace the all the variables with 6, f( 4) = ( 4) 2

+ 2( 4) + 4

f ( 4) = (16) + ( 8) + 4

f ( 4) = 12

Therefore, f( 4) = 12 or simply as ( 4;12) :

Example 3: Given g(x ) = x2

+ 2 x- 1. Find g(2y).

Answer in terms of y.

g(2 y) = x2

+ 2 x 1

g (2 y) = (2 y)2

+ 2(2 y) 1

g (2 y) = 4 y2

+ 4 y 1

Therefore, 4( y)2

+ 4 y 1:

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Example 4: Given

f(x ) = 2 x2

+ 4 x- 12, nd f(2 x+ 4).

Solution:

f(2 x+ 4) = 2 x2

+ 4 x 12

= 2(2 x+ 4) 2

+ 4(2 x+ 4) 12

= 2(2 x+ 4)(2 x+ 4) + 4(2 x+ 4) 12

= 2(4 x2

+ 16 x+ 16) + 4(2 x+ 4) 12

= (8 x2

+ 32 x+ 32) + (8 x+ 16) 12

Combine like terms f(2 x+ 4) = 8 x2

+ (32 x+ 8 x) + (32 + 16 12)

= 8 x2

+ 40 x+ 36

= 2(2 x2

+ 10 x+ 9)

Therefore, f(2 x+ 4) = 2(2 x2

+ 10 x+ 9).

Example 5: Given f(x ) = x2

-x – 4. If f(m ) = 8, compute the value of m

Solution: Make the function f(x ) equivalent to f(m )

x 2

x 4 = 8

x 2

x 12 = 0

( x 4)( x+ 3) = 0

x 4 + 0; x+ 3 = 0

x = 4; x= 3

Therefore, the value of a can be either 4 or -3.

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Exercises:

Evaluate the functions

given:

1. p(x ) = 2x + 1, nd p(-2)

2. p(x ) = 4 x, nd p(-4)

3. g(n ) = 3 n2

+ 6, nd g(8)

4. g(x ) = x3

+ 4 x, nd g(5)

5. f(n ) = n3

+ 3 n2

, nd f(-5)

6. w(a ) = a2

+ 5 a, nd w(7)

7. p(a ) = a3

– 4 a, nd p(-6)

8. f(n ) = 4 3

n

+ 8 5

, nd

f(-1)

9. f(x) = -1 + 1 4

x;

nd f(3 4

)

10. h(n) = n3

+ 6 n, nd h(4)

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Answers in Exercises:

1. 5

2. -16

3. 198

4. 145

5. -50

6. 84

7. -192

8. 4 15

9. – 13 16

10. 88

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Lesson 3: Operations on Functions

Let h(x) and g(x) be functions, and the operations on these two functions is

shown below:

Adding two functions as:

(h+g)(x) = h(x)+g(x)

Subtracting two functions as:

(h-g)(x) = h(x) – g(x)

Multiplying two functions as:

(h g)(x) = h(x) g(c)

Dividing two functions as:

( h g

)(x) = h

(x ) g

(x ) ; whereg

(x ) 6

= 0

Example 1:

Let f(x) = 4x + 5 and g(x) = 3x. Find (f+g)(x), (f-g)(x), (f g)(x), and ( f g

)(x).

(f+g)(x) = (4x+5) + (3x) = 7x+5

(f-g)(x) = (4x+5) – (3x) = x+5

(f g)(x) = (4x+5) (3x) = 12 x2

+5x

(f g

)(x) = 4

x +5 3

x

Example 2:

Let f(x)= 3x+2 and g(x)= 5x-1. Find (f+g)(x), (f-g)(x), (f g)(x), and ( f g

)(x).

(f+g)(x) = (3x+2) + (5x-1) = 8x+1

(f-g)(x) = (3x+2) – (5x-1) = -2x+3

(f g) = (3x+2) (5x-1) = 15 x2

+7x -2

(f g

)(x) = 3

x +2 5

x 1

Example 3:

Let v(x) = x3

and w(x) = 3 x2

+5x. Find (v+w)(x), (v-w)(x), (v w)(x), and

( v w

)(x).

(v+w)(x) = ( x3

) + (3 x2

+5x) = x3

+ 3 x2

+5x

(v-w)(x) = ( x3

) (3×2

+5x) = x3

3x 2

-5x

(v w) = ( x3

) (3×2

+5x) = 3 x5

+ 5 x4

(v w

)(x) = ( x

3 3

x 2

+5 x) = x

x 2 x

(3 x+5) = x

2 3

x +5

Example 4:

Let f(x) = 4 x3

+ 2 x2

+4x + 1 and g(x) = 3 x5

+ 4 x2

+8x-12. Find (f+g)(x),

(f-g)(x), (f g)(x), and ( f g

)(x).

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(f+g)(x) = (4 x3

+ 2 x2

+4x+1) + (3 x5

+ 4 x2

+8x-12) = 3 x5

+ 4 x3

+ 6 x2

+12x

-11

(f-g)(x) = (4 x3

+ 2 x2

+4x+1) – (3 x5

+ 4 x2

+8x-12) = 3x 5

+ 4 x3

2x 2

-4x+13

(f g)(x) = (4 x3

+ 2 x2

+4x+1) (3 x5

+ 4 x2

+8x-12)

= 12 x8

+ 6 x7

+ 12 x6

+ 19 x5

+ 40 x4

16×3

+ 12 x2

40x 12

(f g

)(x) = (4

x3

+2 x2

+4 x+1) (3

x5

+4 x2

+8 x 12)

Example 5:

Let h(x) = 1 and g(x) = x4

x3

+ x2

-1. Find (h+g)(x), (h-g)(x), (h g)(x),

and ( h g

)(x).

(h+g)(x) = (1) + ( x4

x3

+ x2

-1) = x4

x3

+ x2

(h-g)(x) = (1) – ( x4

x3

+ x2

-1) = x4

x3

+ x2

+2

(h g)(x) = (1) (x 4

x3

+ x2

-1) = x4

x3

+ x2

-1

(h g

)(x) = 1 x

4

x3

+ x2

1

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Exercises:

1. If h(x) = 7x+3 and g(x) = 2 x2

+1. Find (f+g)(x)

2. If f(x) = x5

-18 and g(x) = x2

– 6x + 9, what is the vaue of (g-h)(x)?

3. If t(x) = 25 x5

and s(x) = 55 x8

, what is the value of ( t s

)(x)?

4. If v(x) = x3

and w(x) = x2

+ 4, solve (v w)(x)?

5. If f(x) = 4x + 11 and g(x) = 5x + 9, nd (f+g)(x).

6. If f(z) = 7z – 4 and g(z) = z-2, nd (f-g)(x).

7. If f(x) =8 x2

-20 and g(x) =-4, nd( f g

)(x).

8. If f(x) = 2x+2 and g(x) = 9 x2

, what is the value of (f g)(x)?

9. If f(x) = 7 x2

+ 8x -3 and g(x) = 7x, solve for (f g)(x)?

10. If f(x) = 35 x8

– 45x and g(x) = 5x, what is the value of ( f g

)(x).

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Answers to Operations on Functions Exercises:

1. 2 x2

+7x +4

2. x5

x2

+ 6x – 27

3. 5

x 11

x3

4. x5

+ 4 x3

5. 9x +20

6. 6z -2

7. 2x 2

+ 5

8. 18 x3

+ 18 x2

9. 49 x3

+ 56 x2

– 21

10.7 x7

– 9

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