Section 1.6 – Inverse FunctionsSection 1.7 - Logarithms

For an inverse to exist, the function must be one-to-one

(For every x is there is no more than one y)

(For every y is there is no more than one x)

(Must pass BOTH vertical and horizontal line test)

(No x value repeats…no y value repeats)

As with all AP Topics, we will look at it:•Numerically•Graphically•Analytically – Using a methodology involving

algebra or other methods

NUMERICALLY

x 1.1 1.01 1 0.99 0.9y 2.2 2.02 2 1.98 1.8

•Since no x-value has more than one y-value it IS a function.•Since no y-value has more than one x-value it IS one-to-one•THEREFORE, an inverse exists.

x 2.2 2.02 2 1.98 1.8y 1.1 1.01 1 0.99 0.9

represents the inverse of the function

•Domain of the inverse function is 2.2, 2.02, 2, 1.98, 1.8•Range of the inverse function is 1.1, 1.01, 1, 0.99, 0.9

GRAPHICALLY – EXAMPLE #1

f(x) 1f x

GRAPHICALLY – EXAMPLE #2

f(x)

(-2, 2)

(0, 3)

(1, 1)

(3, -1)(-2, -2)

(-3, -3)

1f x

ANALYTICALLY

1Find f x if f x 7x

1. Since f(x) represents a line, it is one-to-one, and thus an inverse exists.2. To find the inverse, switch the x and y, and re-solve for y

f x 7x

1x 7f x

11x f x

7

1Find g x i xf g x 3

1. Is g(x) one-to-one?????

1x g x 3

2 1x g x 3

2 1x 3 g x

Note: The inverse function is suddenly NOT one-to-one

2 1x 3 g x if x 3

WARNING WARNING WARNING

1 2Find g x if g x x 2

21x g x 2

21x 2 g x

1x 2 g x

The original function isNOT one-to-one, so

no inverse exists.

xa Note :y a log a 0 and 0y x y

52 lo 5 225 g5 2

60 lo 016 g 1

2338 8log 2

211l

1 1o

2 2g 2

x1

255

x 2

x 14

2 1

x2

Evaluate: 1/ 5log 25

Evaluate: 4

1log

2

Solve for x: xlog 81 4 4x 81 x 3

Solve for x: 4log x 4 44 x 1

x256

blog c a, b 0, c 0

nAlog logA logB

BlogA nlologAB log B gAA log

Simplify: log6 log5

log30

Simplify: 2 2 2log 12 log 2 log 7

2 2log 6 log 7

2log 42

35 5log x log x 2

5 53log x log x 2

52log x 2

5log x 1x 5

3 3log n log 4 2

3

nlog 2

4

n9 n 36

4

blog c a, b 0, c 0

nAlog logA logB

BlogA nlologAB log B gAA log

6 6 6log n log n 1 log 3

6 6

nlog log 3

n 1

n

3n 1

n 3 n 1

2n 3

3n

2

6log x 1 log x 2 log 6

log x 1 x 2 1 2x x 2 10

2x x 12 0

x 4 x 3 0

x 4, 3 X

blog c a, b 0, c 0

nAlog logA logB

BlogA nlologAB log B gAA log

5logx log x 21 log 25

2log x 21x 2 2x 21x 100

2x 21x 100 0

x 25 x 4 0

x 25, 4 X

22 2log 9t 5 log t 1 2

2 2

9t 5log 2

t 1

2

9t 54

t 1

29t 5 4 t 1 24t 9t 9 0

4t 3 t 3 0

3t , 3

4

X

27log x 1

7 3

2x 1 3

2x 4

x 2

x3 41.23xlog3 log41.23

log41.23x

log3

x 3.386

x 25 9 x 2 log5 log9

log9x 2

log5

log9x 2

log5

x 0.635

4x log 17log17

xlog4

x 2.044

5x log 21log21

xlog5

x 1.892

blog c a, b 0, c 0

nAlog logA logB

BlogA nlologAB log B gAA log