Calculus 1.1.1 Functions

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Presentation Transcript

Calculus 101 : 

Calculus 101

Calculus 101Unit 1: Functions : 

Calculus 101Unit 1: Functions

Calculus 101Unit I Functions : 

Calculus 101Unit I Functions Chapter 1 Definition of a FunctionOne-to-one FunctionsFunction NotationInverse Functions

Slide 4: 

Definition of a Function:

Slide 5: 

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. Definition of a Function:

Slide 6: 

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. Definition of a Function: Example: y=2x+3 X is the input value, Y is the output value

Slide 7: 

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. Definition of a Function: Example: y=2x+3 X is the input value, Y is the output value If x=2, y=7

Slide 8: 

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. Definition of a Function: Example: y=2x+3 X is the input value, Y is the output value If x=2, y=7 If x=-2, y=-1

Slide 9: 

A function associates a single output to each input element, although different inputs may have the same output. Example: y=x2 If x=2, y=4 If x=-2, y=4 Function!

Slide 10: 

x=y2

Slide 11: 

x=y2 X is our input value Y is our output value If x=9, y could equal 3 or -3

Slide 12: 

x=y2 X is our input value Y is our output value If x=9, y could equal 3 or -3 NOT a function!

Slide 13: 

x=y2 X is our input value Y is our output value If x=9, y could equal 3 or -3 NOT a function! Why? There may be only ONE y value for every x value.

Slide 14: 

x=y2 X is our input value Y is our output value If x=9, y could equal 3 or -3 NOT a function! Why? There may be only ONE y value for every x value.

Slide 15: 

y=x2

Slide 16: 

y=x2

Slide 17: 

y=x2

Slide 18: 

y=x2

Slide 19: 

“vertical line test”

Slide 20: 

“vertical line test”

Slide 21: 

“vertical line test”

Slide 22: 

“vertical line test”

Slide 23: 

“vertical line test”

Slide 24: 

x=y2

Slide 25: 

x=y2

Slide 26: 

x=y2

Slide 27: 

x=y2

Slide 28: 

Not a function x=y2

Slide 29: 

One-to-one function

Slide 30: 

One-to-one function For every x value, there is one unique y value

: 

One-to-one function For every x value, there is one unique y value AND For every y value, there is one unique x value

Slide 32: 

One-to-one function For every x value, there is one unique y value AND For every y value, there is one unique x value Every input produces a single, unique output.

Slide 33: 

One-to-one function For every x value, there is one unique y value AND For every y value, there is one unique x value Every input produces a single, unique output. Also known as an injective function

Slide 34: 

For a function to be one-to-one, it must pass the vertical line test AND the horizontal line test

Slide 35: 

For a function to be one-to-one, it must pass the vertical line test AND the horizontal line test

Slide 36: 

For a function to be one-to-one, it must pass the vertical line test AND the horizontal line test

Slide 37: 

For a function to be one-to-one, it must pass the vertical line test AND the horizontal line test

Slide 38: 

y=x3

Slide 39: 

y=x3

Slide 40: 

y=x3 For every input value, there is only one unique output value.

Slide 41: 

y=x3 For every input value, there is only one unique output value.

Slide 42: 

y=x3 For every input value, there is only one unique output value.

Slide 43: 

y=x3 For every input value, there is only one unique output value.

Exercise : 

Exercise

Exercise : 

Exercise Come up with 3 possible one-to-one functions.

Exercise : 

Exercise Come up with 3 possible one-to-one functions. Graph each.

Exercise : 

Exercise Come up with 3 possible one-to-one functions. Graph each. Perform the vertical and horizontal line tests.

Slide 48: 

Function Notation

Slide 49: 

Function Notation Proper function notation replaces the output value, or y, with f(x).

Slide 50: 

Function Notation Proper function notation replaces the output value, or y, with f(x). Indicates how the output value varies in relationship to the input value, x.

Slide 51: 

Function Notation Proper function notation replaces the output value, or y, with f(x). Indicates how the output value varies in relationship to the input value, x. Indicates there is only one output value for every input value.

Slide 52: 

Example y=x2

Slide 53: 

Example y=x2 f(x)=x2

Slide 54: 

Example y=x2 f(x)=x2 f(-2)=4

Slide 55: 

Example y=x2 f(x)=x2 f(-2)=4 f(5)=25 f(3)=9

Slide 56: 

Example y=x2 f(x)=x2 f(-2)=4 f(5)=25 f(3)=9 1=x2+y2

Slide 57: 

f(x) is the most commonly used function notation

Slide 58: 

f(x) is the most commonly used function notation g(x) b(x) h(x) j(x)

Slide 59: 

f(x) is the most commonly used function notation g(x) b(x) h(x) j(x) f(t) f(m)

Slide 60: 

f(x) is the most commonly used function notation g(x) b(x) h(x) j(x) f(t) f(m) d(t) v(t) h(a)

Slide 61: 

Inverse Functions

Slide 62: 

Inverse Functions An inverse function reverses, or “undoes” the operations of the original function

Slide 63: 

Inverse Functions An inverse function reverses, or “undoes” the operations of the original function Example: f(x)=2x

Slide 64: 

Inverse Functions An inverse function reverses, or “undoes” the operations of the original function Example: f(x)=2x f-1(x)=

Slide 65: 

Inverse Functions An inverse function reverses, or “undoes” the operations of the original function Example: f(x)=2x f-1(x)=

Slide 66: 

Inverse Functions An inverse function reverses, or “undoes” the operations of the original function Example: f(x)=2x f-1(x)= f(-3)= -6

Slide 67: 

Inverse Functions An inverse function reverses, or “undoes” the operations of the original function Example: f(x)=2x f-1(x)= f(-3)= -6 f-1(-6)= -3

Slide 68: 

Inverse Functions An inverse function reverses, or “undoes” the operations of the original function Example: f(x)=2x f-1(x)= f(-3)= -6 f-1(-6)= -3 Example: f(x)=-3x+5

Slide 69: 

Inverse Functions An inverse function reverses, or “undoes” the operations of the original function Example: f(x)=2x f-1(x)= f(-3)= -6 f-1(-6)= -3 Example: f(x)=-3x+5 f-1(x)=

Slide 70: 

Inverse Functions An inverse function reverses, or “undoes” the operations of the original function Example: f(x)=2x f-1(x)= f(-3)= -6 f-1(-6)= -3 Example: f(x)=-3x+5 f-1(x)= f(2)=-3(2)+5

Slide 71: 

Inverse Functions An inverse function reverses, or “undoes” the operations of the original function Example: f(x)=2x f-1(x)= f(-3)= -6 f-1(-6)= -3 Example: f(x)=-3x+5 f-1(x)= f(2)=-3(2)+5 f(2)=-1

Slide 72: 

Inverse Functions An inverse function reverses, or “undoes” the operations of the original function Example: f(x)=2x f-1(x)= f(-3)= -6 f-1(-6)= -3 Example: f(x)=-3x+5 f-1(x)= f(2)=-3(2)+5 f(2)=-1 f-1(-1)=

Slide 73: 

Inverse Functions An inverse function reverses, or “undoes” the operations of the original function Example: f(x)=2x f-1(x)= f(-3)= -6 f-1(-6)= -3 Example: f(x)=-3x+5 f-1(x)= f(2)=-3(2)+5 f(2)=-1 f-1(-1)= f-1(x)=2

Slide 74: 

Finding the inverse of a function

Slide 75: 

Finding the inverse of a function

Slide 76: 

Finding the inverse of a function

Slide 77: 

Finding the inverse of a function

Slide 78: 

Finding the inverse of a function

Slide 79: 

Finding the inverse of a function

Slide 80: 

Finding the inverse of a function

Slide 81: 

Finding the inverse of a function

Slide 82: 

Finding the inverse of a function

Conclusion : 

Conclusion

Slide 94: 

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. Definition of a Function:

Slide 95: 

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. Definition of a Function: A function associates a single output to each input element, although different inputs may have the same output.

Slide 96: 

“vertical line test”

Slide 97: 

One-to-one function For every x value, there is one unique y value AND For every y value, there is one unique x value Every input produces a single, unique output. Also known as an injective function

Slide 98: 

y=x3 For every input value, there is only one unique output value.

Slide 99: 

f(x) is the most commonly used function notation g(x) b(x) h(x) j(x) f(t) f(m) d(t) v(t) h(a)

Slide 100: 

Inverse Functions An inverse function reverses, or “undoes” the operations of the original function Example: f(x)=2x f-1(x)=

Slide 101: 

Finding the inverse of a function

Slide 102: 

Finding the inverse of a function

Slide 103: 

Written and Produced by James Snyder ?2009