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