7.1 – Operations on Functions

Operation Definition

Operation Definition

Sum

Operation Definition

Sum (f + g)(x)

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Difference

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Difference (f – g)(x) =

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Difference (f – g)(x) = f(x) – g(x)

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Difference (f – g)(x) = f(x) – g(x)

Product

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Difference (f – g)(x) = f(x) – g(x)

Product (f · g)(x) =

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Difference (f – g)(x) = f(x) – g(x)

Product (f · g)(x) = f(x) · g(x)

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Difference (f – g)(x) = f(x) – g(x)

Product (f · g)(x) = f(x) · g(x)

Quotient f (x) =

g

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Difference (f – g)(x) = f(x) – g(x)

Product (f · g)(x) = f(x) · g(x)

Quotient f (x) = f(x)

g g(x)

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x)

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

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

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

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

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

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

= (2x – 3)

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

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

= (2x – 3)

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

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

= (2x – 3) + (4x + 9)

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

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

= (2x – 3) + (4x + 9)

= 6x + 6

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

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

= (2x – 3) + (4x + 9)

= 6x – 6

(f – g)(x)

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

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

= (2x – 3) + (4x + 9)

= 6x – 6

(f – g)(x) = f(x) – g(x)

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

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

= (2x – 3) + (4x + 9)

= 6x – 6

(f – g)(x) = f(x) – g(x)

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

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

= (2x – 3) + (4x + 9)

= 6x – 6

(f – g)(x) = f(x) – g(x)

= (2x – 3)

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

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

= (2x – 3) + (4x + 9)

= 6x – 6

(f – g)(x) = f(x) – g(x)

= (2x – 3)

Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for

f(x) gand g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)= (2x – 3) + (4x + 9)= 6x – 6

(f – g)(x) = f(x) – g(x)= (2x – 3) – (4x + 9)

Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for

f(x) gand g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)= (2x – 3) + (4x + 9)= 6x – 6

(f – g)(x) = f(x) – g(x)= (2x – 3) – (4x + 9)

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

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

= (2x – 3) + (4x + 9)

= 6x – 6

(f – g)(x) = f(x) – g(x)

= (2x – 3) – (4x + 9)

= 2x – 3 – 4x

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

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

= (2x – 3) + (4x + 9)

= 6x – 6

(f – g)(x) = f(x) – g(x)

= (2x – 3) – (4x + 9)

= 2x – 3 – 4x – 9

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

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

= (2x – 3) + (4x + 9)

= 6x – 6

(f – g)(x) = f(x) – g(x)

= (2x – 3) – (4x + 9)

= 2x – 3 – 4x – 9

= -2x – 12

(f · g)(x)

(f · g)(x) = f(x) · g(x)

(f · g)(x) = f(x) · g(x)

(f · g)(x) = f(x) · g(x)

= (2x – 3)

(f · g)(x) = f(x) · g(x)

= (2x – 3)

(f · g)(x) = f(x) · g(x)

= (2x – 3)(4x + 9)

(f · g)(x) = f(x) · g(x)

= (2x – 3)(4x + 9)

= 8x2 + 18x – 12x – 27

(f · g)(x) = f(x) · g(x)

= (2x – 3)(4x + 9)

= 8x2 + 18x – 12x – 27

= 8x2 + 6x – 27

(f · g)(x) = f(x) · g(x)

= (2x – 3)(4x + 9)

= 8x2 + 18x – 12x – 27

= 8x2 + 6x – 27

f (x)

g

(f · g)(x) = f(x) · g(x)

= (2x – 3)(4x + 9)

= 8x2 + 18x – 12x – 27

= 8x2 + 6x – 27

f (x) = f(x)

g g(x)

(f · g)(x) = f(x) · g(x)

= (2x – 3)(4x + 9)

= 8x2 + 18x – 12x – 27

= 8x2 + 6x – 27

f (x) = f(x)

g g(x)

= 2x – 3

4x + 9

(f · g)(x) = f(x) · g(x)

= (2x – 3)(4x + 9)

= 8x2 + 18x – 12x – 27

= 8x2 + 6x – 27

f (x) = f(x)

g g(x)

= 2x – 3

4x + 9

*Factor & Simplify if possible!

Composite Function

Composite Function

- taking the function

Composite Function

- taking the function of a function

Composite Function

- taking the function of a function

[f °g(x)]

Composite Function

- taking the function of a function

[f °g(x)] = f[g(x)]

Composite Function

- taking the function of a function

[f °g(x)] = f[g(x)]

Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x2 + x – 1.

Composite Function

- taking the function of a function

[f °g(x)] = f[g(x)]

Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x2 + x – 1.

[f °g(x)] = f[g(x)]

Composite Function

- taking the function of a function

[f °g(x)] = f[g(x)]

Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x2 + x – 1.

[f °g(x)] = f[g(x)]

Composite Function

- taking the function of a function

[f °g(x)] = f[g(x)]

Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x2 + x – 1.

[f °g(x)] = f[g(x)]

= f[x2 + x – 1]

Composite Function

- taking the function of a function

[f °g(x)] = f[g(x)]

Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x2 + x – 1.

[f °g(x)] = f[g(x)]

= f[x2 + x – 1]

Composite Function

- taking the function of a function

[f °g(x)] = f[g(x)]

Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x2 + x – 1.

[f °g(x)] = f[g(x)]

= f[x2 + x – 1]

Composite Function- taking the function of a function

[f °g(x)] = f[g(x)]

Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x2 + x – 1.

[f °g(x)] = f[g(x)]= f(x2 + x – 1)= (x2 + x – 1) + 3

Composite Function- taking the function of a function

[f °g(x)] = f[g(x)]

Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x2 + x – 1.

[f °g(x)] = f[g(x)]= f(x2 + x – 1)= (x2 + x – 1) + 3= x2 + x + 2

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)]

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

= (x + 3)2

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

= (x + 3)2

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

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

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

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

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

= (x + 3)2 + (x + 3) – 1

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

= (x + 3)2 + (x + 3) – 1

= (x + 3)(x + 3) + (x + 3) – 1

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

= (x + 3)2 + (x + 3) – 1

= (x + 3)(x + 3) + (x + 3) – 1

= x2 + 6x + 9 + x + 3 – 1

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

= (x + 3)2 + (x + 3) – 1

= (x + 3)(x + 3) + (x + 3) – 1

= x2 + 6x + 9 + x + 3 – 1

= x2 + 7x + 11

f(x) = x + 3 and g(x) = x2 + x – 1 [g°f(x)] = g[f(x)]

= g(x + 3)= (x + 3)2 + (x + 3) – 1= (x + 3)(x + 3) + (x + 3) – 1= x2 + 6x + 9 + x + 3 – 1 = x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)].

f(x) = x + 3 and g(x) = x2 + x – 1 [g°f(x)] = g[f(x)]

= g(x + 3)= (x + 3)2 + (x + 3) – 1= (x + 3)(x + 3) + (x + 3) – 1= x2 + 6x + 9 + x + 3 – 1 = x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)].

g[f(5)] =

f(x) = x + 3 and g(x) = x2 + x – 1 [g°f(x)] = g[f(x)]

= g(x + 3)= (x + 3)2 + (x + 3) – 1= (x + 3)(x + 3) + (x + 3) – 1= x2 + 6x + 9 + x + 3 – 1 = x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)].

g[f(5)] =

f(x) = x + 3 and g(x) = x2 + x – 1 [g°f(x)] = g[f(x)]

= g(x + 3)= (x + 3)2 + (x + 3) – 1= (x + 3)(x + 3) + (x + 3) – 1= x2 + 6x + 9 + x + 3 – 1 = x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)].

g[f(5)] =

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

= (x + 3)2 + (x + 3) – 1

= (x + 3)(x + 3) + (x + 3) – 1

= x2 + 6x + 9 + x + 3 – 1

= x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)].

g[f(5)] = g[4(5)]

f(x) = x + 3 and g(x) = x2 + x – 1 [g°f(x)] = g[f(x)]

= g(x + 3)= (x + 3)2 + (x + 3) – 1= (x + 3)(x + 3) + (x + 3) – 1= x2 + 6x + 9 + x + 3 – 1 = x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] = g[4(5)]

= g(20)

f(x) = x + 3 and g(x) = x2 + x – 1 [g°f(x)] = g[f(x)]

= g(x + 3)= (x + 3)2 + (x + 3) – 1= (x + 3)(x + 3) + (x + 3) – 1= x2 + 6x + 9 + x + 3 – 1 = x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] = g[4(5)]

= g(20)

f(x) = x + 3 and g(x) = x2 + x – 1 [g°f(x)] = g[f(x)]

= g(x + 3)= (x + 3)2 + (x + 3) – 1= (x + 3)(x + 3) + (x + 3) – 1= x2 + 6x + 9 + x + 3 – 1 = x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] = g[4(5)]

= g(20) = 2(20) – 1

f(x) = x + 3 and g(x) = x2 + x – 1 [g°f(x)] = g[f(x)]

= g(x + 3)= (x + 3)2 + (x + 3) – 1= (x + 3)(x + 3) + (x + 3) – 1= x2 + 6x + 9 + x + 3 – 1 = x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] = g[4(5)]

= g(20) = 2(20) – 1 = 39

7.3 – Square Root Functions & Inequalities

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

x + 4 = 0

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

x + 4 = 0

x = -4

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

x + 4 = 0

x = -4

Domain: { x | x > -4}

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

x + 4 = 0

x = -4

Domain: { x | x > -4}

y = √ x + 4

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

x + 4 = 0

x = -4

Domain: { x | x > -4}

y = √ x + 4

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

x + 4 = 0

x = -4

Domain: { x | x > -4}

y = √ x + 4

y = √ -4+ 4

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

x + 4 = 0

x = -4

Domain: { x | x > -4}

y = √ x + 4

y = √ -4+ 4

y = 0

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

x + 4 = 0

x = -4

Domain: { x | x > -4}

y = √ x + 4

y = √ -4+ 4

y = 0

Range: { y | y > 0}

Ex. 2 Graph each function. State the domain & range.

a. y = √ x + 4

Domain: { x | x > -4}, Range: { y | y > 0}

Graph: Y=

2nd, x2

x + 4)

Zoom:6

2nd Graph

Plot at least 3 points of curve

(x & y ints. & one other pt.)

x y

-4 0

-3 1

0 2

Ex. 3 Graph each inequality

a. y <√ x + 4

Graph: Y= Cursor left to \

Press “Enter” until

(If > make it )

2nd, x2

x + 4)

Zoom:6

2nd Graph

Plot at least 3 points of curve

(x & y ints. & one other pt.)

x y

-4 0

-3 1

0 2