Calculus 1.1.2 Functions

Calculus 101 : 

Calculus 101

Calculus 101Unit I: Functions : 

Calculus 101Unit I: Functions Chapter 2 Composite Functions Properties of Inverse Functions Odd and Even Functions

Composite Functions : 

Composite Functions

Composite Functions : 

Composite Functions g(x) b(x) h(x) j(x) f(t) f(m) d(t) v(t) h(a)

Composite Functions : 

Composite Functions let f(x)=2x

Composite Functions : 

Composite Functions let f(x)=2x let g(x)=3+x

Composite Functions : 

Composite Functions let f(x)=2x let g(x)=3+x f(x)

Composite Functions : 

Composite Functions let f(x)=2x let g(x)=3+x f(g(x))=?

Composite Functions : 

Composite Functions let f(x)=2x let g(x)=3+x f(g(x))=2(x)

Composite Functions : 

Composite Functions let f(x)=2x let g(x)=3+x f(g(x))=2(3+x)

Composite Functions : 

Composite Functions let f(x)=2x let g(x)=3+x f(g(x))=2(3+x) g(f(x))=?

Composite Functions : 

Composite Functions let f(x)=2x let g(x)=3+x f(g(x))=2(3+x) g(f(x))=3+x

Composite Functions : 

Composite Functions let f(x)=2x let g(x)=3+x f(g(x))=2(3+x) g(f(x))=3+2x

Composite Functions : 

Composite Functions let f(x)=2x let g(x)=3+x f(g(x))=2(3+x) g(f(x))=3+2x g(g(x))=?

Composite Functions : 

Composite Functions let f(x)=2x let g(x)=3+x f(g(x))=2(3+x) g(f(x))=3+2x g(g(x))=3+x

Composite Functions : 

Composite Functions let f(x)=2x let g(x)=3+x f(g(x))=2(3+x) g(f(x))=3+2x g(g(x))=3+(3+x)

Composite Functions : 

Composite Functions let f(x)=2x let g(x)=3+x f(g(x))=2(3+x) g(f(x))=3+2x g(g(x))=6+x

Composite Functions : 

Composite Functions let f(x)=3x2 let h(x)= let g(x)=2x-4

Composite Functions : 

Composite Functions let f(x)=3x2 let h(x)= let g(x)=2x-4 f(g(h(x)))=?

Composite Functions : 

Composite Functions let f(x)=3x2 let h(x)= let g(x)=2x-4 f(g(h(x)))=3x2

Composite Functions : 

Composite Functions let f(x)=3x2 let h(x)= let g(x)=2x-4 f(g(h(x)))=3(2x-4)2

Composite Functions : 

Composite Functions let f(x)=3x2 let h(x)= let g(x)=2x-4 f(g(h(x)))=3(2 -4)2

Composite Functions : 

Composite Functions let f(x)=3x2 let h(x)= let g(x)=2x-4 h(g(f(x)))=?

Composite Functions : 

Composite Functions let f(x)=3x2 let h(x)= let g(x)=2x-4 h(g(f(x)))=

Composite Functions : 

Composite Functions let f(x)=3x2 let h(x)= let g(x)=2x-4 h(g(f(x)))=

Composite Functions : 

Composite Functions let f(x)=3x2 let h(x)= let g(x)=2x-4 h(g(f(x)))=

Properties of Inverse Functions : 

Properties of Inverse Functions

Properties of Inverse Functions : 

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

Properties of Inverse Functions : 

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

Properties of Inverse Functions : 

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

Properties of Inverse Functions : 

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

Properties of Inverse Functions : 

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

Properties of Inverse Functions : 

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

Properties of Inverse Functions : 

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

Properties of Inverse Functions : 

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

Properties of Inverse Functions : 

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

Properties of Inverse Functions : 

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

Properties of Inverse Functions : 

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

Properties of Inverse Functions : 

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

Properties of Inverse Functions : 

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

Properties of Inverse Functions : 

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

Properties of Inverse Functions : 

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

Properties of Inverse Functions : 

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

Properties of Inverse Functions : 

Properties of Inverse Functions

Properties of Inverse Functions : 

Properties of Inverse Functions

Properties of Inverse Functions : 

Properties of Inverse Functions

Properties of Inverse Functions : 

Properties of Inverse Functions y=x

Properties of Inverse Functions? : 

Properties of Inverse Functions? y=x

Inverse Functions : 

Inverse Functions Do all functions have an inverse?

Inverse Functions : 

Inverse Functions Do all functions have an inverse?

Inverse Functions : 

Inverse Functions Do all functions have an inverse?

Inverse Functions : 

Inverse Functions Do all functions have an inverse? not all functions have an inverse

Inverse Functions : 

Inverse Functions one-to-one function

Inverse Functions : 

Inverse Functions one-to-one function

Inverse Functions : 

Inverse Functions one-to-one function

Inverse Functions : 

Inverse Functions one-to-one function

Properties of Inverse Functions : 

Properties of Inverse Functions

Properties of Inverse Functions : 

Properties of Inverse Functions if a function is a one-to-one function, it has an inverse function that is also one-to-one.

Properties of Inverse Functions : 

Properties of Inverse Functions if a function is a one-to-one function, it has an inverse function that is also one-to-one. if a function is not a one-to-one function, it does not have an inverse

Properties of Inverse Functions : 

Properties of Inverse Functions if a function is a one-to-one function, it has an inverse function that is also one-to-one. if a function is not a one-to-one function, it does not have an inverse inverse functions are symmetrical about the line y=x

Properties of Inverse Functions : 

Properties of Inverse Functions if a function is a one-to-one function, it has an inverse function that is also one-to-one if a function is not a one-to-one function, it does not have an inverse all inverse functions are symmetrical about the line y=x if a composite function is created out of two inverse functions, the result is always x.

Odd and Even Functions : 

Odd and Even Functions

Odd and Even Functions : 

Odd and Even Functions In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses.

Even Functions : 

Even Functions

Even Functions : 

Even Functions

Even Functions : 

Even Functions the output value for a function is the same for both an input value and the opposite of that input value

Even Functions : 

Even Functions

Even Functions : 

Even Functions

Even Functions : 

Even Functions

Odd Functions : 

Odd Functions

Odd Functions : 

Odd Functions the opposite of a function with a given input value will yield the same as a function with the opposite input value

Odd Functions : 

Odd Functions

Odd Functions : 

Odd Functions

Odd Functions : 

Odd Functions

Odd Functions : 

Odd Functions

Odd and Even Functions : 

Odd and Even Functions

Odd and Even Functions : 

Odd and Even Functions even functions are symmetrical about the y-axis

Odd and Even Functions : 

Odd and Even Functions

Odd and Even Functions : 

Odd and Even Functions odd functions are symmetrical about the origin, at a 180-degree rotation

Slide 80: 

In conclusion…

Composite Functions : 

Composite Functions let f(x)=2x let g(x)=3+x f(g(x))=2(3+x) g(f(x))=3+2x

Properties of Inverse Functions : 

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

Inverse Functions : 

Inverse Functions one-to-one function

Odd Functions : 

Odd Functions Even Functions

Slide 85: 

Written and Produced by James Snyder ?2009