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------=_NextPart_01C9141E.1361F110 Content-Location: file:///C:/C95ACA62/CalcCh1Sec5Part11B_files/master07.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C9141E.1361F110 Content-Location: file:///C:/C95ACA62/CalcCh1Sec5Part11B_files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C9141E.1361F110 Content-Location: file:///C:/C95ACA62/CalcCh1Sec5Part11B_files/slide0001.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Ch 1 Sec 5 Part 1
1.5 Functions and Logarithms
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file:///C:/C95ACA62/CalcCh1Sec5Part11B_files/slide0002.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Ch 1 Sec 5 Part 1
Goals
nTo identify a one-to-one function&#= 13;
nTo determine the algebraic representation and the graphical representation of a <= span style=3D'position:absolute;top:67.85%;left:3.64%;width:82.14%;height:32.14= %'>function and its inverse
nTo apply properties of logarithms&#= 13;
nTo use logarithmic regression to so= lve = problems
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One-to-One Functions
nA function f(x= ) is one-to-one on a domain D if f(a) ≠ f(b) whenever a ≠ b.
<= span style=3D'visibility:hidden'>n
nHow do we know if a function is on= e-to-one?
<= span class=3DBB style=3D'position:absolute;left:-4.43%;top:.3em'>nEasiest way…graph it!
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Horizontal Line Test
nThe graph of a one-to-one function y =3D = f(x) can intersect any horizontal line no more than one time. nIf it intersects the graph more than once, then two different x values have the same y value and therefore is n= ot one-to-one.
<= span class=3DBB style=3D'position:absolute;left:-4.43%;top:.3em'>n(One-t= o-one: one x, one y).
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Example 1
nGraph=
n
<= span class=3DHBB style=3D'position:absolute;left:-4.36%;top:.3em'>nWhich= of these functions are one-to-one?
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Example 1
nGraph=
n
n
n
n
n
n
<= span class=3DHBB style=3D'position:absolute;left:-4.36%;top:.3em'>nWhich= of these functions are one-to-one?
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Example 1
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Inverses
nSince every output of a one-to-one function comes from only one input, we can reverse the = function to send the outputs back to the inputs = from which they came. = nThe inverse of f is defined by reversing a one-to-one function f. nThe domain and rang= e of inverse functions are reverse= d (or switched or transposed)
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Notation
<= span class=3DBB style=3D'position:absolute;left:-3.69%;top:.3em'>nThe sy= mbol for the inverse of f is f -1, which is read, “f inverse.”
<= span class=3DBB style=3D'position:absolute;left:-4.31%;top:.3em'>nImport= ant note: the -1 in f -1 is not an exponent
¨= [f -1(x) does not mean (1/f(x))
n
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Compositions
nf(x) =3D x3 ; g(x) =3D x1/3 = ;
nFind f(g(x)) and g(f(x))&= #13;
<= span style=3D'visibility:hidden'>n
<= span style=3D'visibility:hidden'>n
nf(x) =3D x ; g(x) =3D 1/x= 3;
nFind f(g(x)) and g(f(x))<= /i>
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Inverse Functions
nThe result of com= posing a function and its inverse in either order is the identity = function, which is the function that assigns each number to itself.
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Finding Inverses
<= span class=3DBB style=3D'position:absolute;left:-4.43%;top:.3em'>nTo fin= d the inverse of a function f
<= span class=3DB1B style=3D'position:absolute;left:-3.21%;top:.22em'>¨= Solve the equation y =3D f(x) for x in terms of y.
<= span class=3DB1B style=3D'position:absolute;left:-3.13%;top:.22em'>¨= Interchange x and y. The resulting formula will be y =3D f -1(x).