7.1 – operations on functions
DESCRIPTION
7.1 – Operations on Functions. OperationDefinition. OperationDefinition Sum. OperationDefinition Sum( f + g )( x ). OperationDefinition Sum( f + g )( x ) = f ( x ) + g ( x ). OperationDefinition Sum( f + g )( x ) = f ( x ) + g ( x ) - PowerPoint PPT PresentationTRANSCRIPT
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