Functions and Logarithms

One-to-One FunctionsA function f(x) is one-to-one if

f(a) ≠ f(b) whenever a ≠ b.Must pass horizontal line test.

Not one-to-oneOne-to-one

InversesIf a function is one-to-one, then it

has an inverse.◦Notation: f-1(x)◦IMPORTANT:

!!!!!)(

1meannot does )(1

xfxf

InversesTo find the inverse of a function,

solve for x, “swap” x and y, and solve for y.

Ex: What is the inverse of f(x) = -2x + 4?

y = -2x + 4y – 4 = -2x

xy

2

4

yx

2

4(“swap” x and y)

yx 22

1

Therefore,

22

1)(1 xxf

InversesInverse functions are symmetric

(reflected) about the line y = x.Therefore, in order for two

functions to be inverses, the results of the composites is x.

x x f f x x f f

)) ( ( AND )) ( (1 1

InversesExample: Prove the two

functions from our last example are inverses. 2

2

1)( and 42)( 1

xxfxxf

22

1))(( 1 xfxff

422

12

x

422

12

x

44 x

= x

Now, we must check the other composite!

42))(( 11 xfxff

2)42(2

1

x

22 x

= xTherefore, these two functions must be inverses of one another!!!

Logarithmsy = logax means ay = x

◦Ex: log381 = 4 means 34 = 81

What is the inverse of y = logax?Since y = logax is ay = x, then the inverse has to beax = y

LogarithmsCommon logarithms:

log x means log10x

ln x means logex

Inverse Properties of Logs

xa xa log xa xa log

(both of these hold true if a > 1 and x > 0)

xe x ln xex ln(both of these hold true if x > 0)

Inverse Properties of LogsExample:

Solve ln x = 3t + 5 for x.

53ln tx53ln tx ee (use each side as an exponent

of e)53 tex (e and ln are inverses and “undo”

each other.)

Inverse Property of LogsExample:

Solve e2x = 10 for x.

102 xe10lnln 2 xe (take the natural log of both

sides)

10ln2 x (ln and e are inverses and “undo” each other.)

151.12

10lnx

Properties of LogarithmsFor any real numbers x > 0 and y

> 0,xyyx aaa logloglog

y

xyx aaa logloglog

xyx ay

a loglog

Change of Base PropertySince our calculators will not

calculate logs of bases other than 10 or e,

a

xxa ln

lnlog

ExampleSarah invests $1000 in an account that

earns 5.25% interest compounded annually. How long will it take the account to reach $2500?t)0525.01(10002500

t)0525.1(5.2 (divide both sides by 1000)

t)0525.1ln()5.2ln( (take a log of both sides…doesn’t matter what base you use!!!)

)0525.1ln()5.2ln( t (by my property, exponent comes out front)

)0525.1ln(

)5.2ln(t (divide by ln(1.0525))t ≈ 17.907

years