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
![Page 1: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/1.jpg)
7.1 – Operations on Functions
![Page 2: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/2.jpg)
Operation Definition
![Page 3: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/3.jpg)
Operation Definition
Sum
![Page 4: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/4.jpg)
Operation Definition
Sum (f + g)(x)
![Page 5: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/5.jpg)
Operation Definition
Sum (f + g)(x) = f(x) + g(x)
![Page 6: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/6.jpg)
Operation Definition
Sum (f + g)(x) = f(x) + g(x)
Difference
![Page 7: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/7.jpg)
Operation Definition
Sum (f + g)(x) = f(x) + g(x)
Difference (f – g)(x) =
![Page 8: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/8.jpg)
Operation Definition
Sum (f + g)(x) = f(x) + g(x)
Difference (f – g)(x) = f(x) – g(x)
![Page 9: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/9.jpg)
Operation Definition
Sum (f + g)(x) = f(x) + g(x)
Difference (f – g)(x) = f(x) – g(x)
Product
![Page 10: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/10.jpg)
Operation Definition
Sum (f + g)(x) = f(x) + g(x)
Difference (f – g)(x) = f(x) – g(x)
Product (f · g)(x) =
![Page 11: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/11.jpg)
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)
![Page 12: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/12.jpg)
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
![Page 13: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/13.jpg)
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)
![Page 14: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/14.jpg)
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
![Page 15: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/15.jpg)
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)
![Page 16: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/16.jpg)
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)
![Page 17: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/17.jpg)
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)
![Page 18: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/18.jpg)
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)
![Page 19: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/19.jpg)
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)
![Page 20: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/20.jpg)
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)
![Page 21: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/21.jpg)
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
![Page 22: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/22.jpg)
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)
![Page 23: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/23.jpg)
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)
![Page 24: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/24.jpg)
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)
![Page 25: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/25.jpg)
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)
![Page 26: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/26.jpg)
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)
![Page 27: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/27.jpg)
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)
![Page 28: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/28.jpg)
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)
![Page 29: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/29.jpg)
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
![Page 30: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/30.jpg)
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
![Page 31: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/31.jpg)
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
![Page 32: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/32.jpg)
(f · g)(x)
![Page 33: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/33.jpg)
(f · g)(x) = f(x) · g(x)
![Page 34: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/34.jpg)
(f · g)(x) = f(x) · g(x)
![Page 35: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/35.jpg)
(f · g)(x) = f(x) · g(x)
= (2x – 3)
![Page 36: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/36.jpg)
(f · g)(x) = f(x) · g(x)
= (2x – 3)
![Page 37: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/37.jpg)
(f · g)(x) = f(x) · g(x)
= (2x – 3)(4x + 9)
![Page 38: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/38.jpg)
(f · g)(x) = f(x) · g(x)
= (2x – 3)(4x + 9)
= 8x2 + 18x – 12x – 27
![Page 39: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/39.jpg)
(f · g)(x) = f(x) · g(x)
= (2x – 3)(4x + 9)
= 8x2 + 18x – 12x – 27
= 8x2 + 6x – 27
![Page 40: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/40.jpg)
(f · g)(x) = f(x) · g(x)
= (2x – 3)(4x + 9)
= 8x2 + 18x – 12x – 27
= 8x2 + 6x – 27
f (x)
g
![Page 41: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/41.jpg)
(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)
![Page 42: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/42.jpg)
(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
![Page 43: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/43.jpg)
(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!
![Page 44: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/44.jpg)
Composite Function
![Page 45: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/45.jpg)
Composite Function
- taking the function
![Page 46: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/46.jpg)
Composite Function
- taking the function of a function
![Page 47: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/47.jpg)
Composite Function
- taking the function of a function
[f °g(x)]
![Page 48: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/48.jpg)
Composite Function
- taking the function of a function
[f °g(x)] = f[g(x)]
![Page 49: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/49.jpg)
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.
![Page 50: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/50.jpg)
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)]
![Page 51: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/51.jpg)
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)]
![Page 52: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/52.jpg)
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]
![Page 53: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/53.jpg)
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]
![Page 54: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/54.jpg)
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]
![Page 55: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/55.jpg)
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
![Page 56: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/56.jpg)
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
![Page 57: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/57.jpg)
f(x) = x + 3 and g(x) = x2 + x – 1
[g°f(x)]
![Page 58: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/58.jpg)
f(x) = x + 3 and g(x) = x2 + x – 1
[g°f(x)] = g[f(x)]
![Page 59: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/59.jpg)
f(x) = x + 3 and g(x) = x2 + x – 1
[g°f(x)] = g[f(x)]
![Page 60: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/60.jpg)
f(x) = x + 3 and g(x) = x2 + x – 1
[g°f(x)] = g[f(x)]
![Page 61: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/61.jpg)
f(x) = x + 3 and g(x) = x2 + x – 1
[g°f(x)] = g[f(x)]
= g(x + 3)
![Page 62: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/62.jpg)
f(x) = x + 3 and g(x) = x2 + x – 1
[g°f(x)] = g[f(x)]
= g(x + 3)
![Page 63: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/63.jpg)
f(x) = x + 3 and g(x) = x2 + x – 1
[g°f(x)] = g[f(x)]
= g(x + 3)
![Page 64: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/64.jpg)
f(x) = x + 3 and g(x) = x2 + x – 1
[g°f(x)] = g[f(x)]
= g(x + 3)
= (x + 3)2
![Page 65: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/65.jpg)
f(x) = x + 3 and g(x) = x2 + x – 1
[g°f(x)] = g[f(x)]
= g(x + 3)
= (x + 3)2
![Page 66: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/66.jpg)
f(x) = x + 3 and g(x) = x2 + x – 1
[g°f(x)] = g[f(x)]
= g(x + 3)
= (x + 3)2 + (x + 3)
![Page 67: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/67.jpg)
f(x) = x + 3 and g(x) = x2 + x – 1
[g°f(x)] = g[f(x)]
= g(x + 3)
= (x + 3)2 + (x + 3)
![Page 68: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/68.jpg)
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
![Page 69: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/69.jpg)
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
![Page 70: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/70.jpg)
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
![Page 71: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/71.jpg)
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
![Page 72: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/72.jpg)
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)].
![Page 73: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/73.jpg)
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)] =
![Page 74: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/74.jpg)
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)] =
![Page 75: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/75.jpg)
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)] =
![Page 76: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/76.jpg)
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)]
![Page 77: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/77.jpg)
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)
![Page 78: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/78.jpg)
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)
![Page 79: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/79.jpg)
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
![Page 80: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/80.jpg)
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
![Page 81: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/81.jpg)
7.3 – Square Root Functions & Inequalities
![Page 82: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/82.jpg)
Ex. 1 Identify the domain & range of each function.
a. y = √ x + 4
![Page 83: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/83.jpg)
Ex. 1 Identify the domain & range of each function.
a. y = √ x + 4
x + 4 = 0
![Page 84: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/84.jpg)
Ex. 1 Identify the domain & range of each function.
a. y = √ x + 4
x + 4 = 0
x = -4
![Page 85: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/85.jpg)
Ex. 1 Identify the domain & range of each function.
a. y = √ x + 4
x + 4 = 0
x = -4
Domain: { x | x > -4}
![Page 86: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/86.jpg)
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
![Page 87: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/87.jpg)
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
![Page 88: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/88.jpg)
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
![Page 89: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/89.jpg)
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
![Page 90: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/90.jpg)
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}
![Page 91: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/91.jpg)
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.)
![Page 92: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/92.jpg)
x y
-4 0
-3 1
0 2
![Page 93: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/93.jpg)
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.)
![Page 94: 7.1 – Operations on Functions](https://reader031.vdocument.in/reader031/viewer/2022033102/56814f9a550346895dbd57a1/html5/thumbnails/94.jpg)
x y
-4 0
-3 1
0 2