MAC 1105-Inverse Functions(Gohar)Sec PDF

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MAC 1105-Inverse Functions(Gohar)Sec...

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Inverse Functions

MAC 1105

One-to-One Function A function f(x) is one-to-one for a and b in the domain of f, or

EXAMPLE

1a. Determine if the relation is one-to-one.

YOUR TURN

1b. Determine if the relation is one-to-one.

{( 2,5) ,( 4, −2) ,( 7,5) ,( −2,1)}

{( 3,9) ,( 4,16) ,( −7, 49) ,( −3,9)}

Horizontal Line Test A function than one point.

is one-to-one only if any horizontal line intersects the graph at no more

MDC Mathematics 1

EXAMPLE

YOUR TURN

2a. Determine whether each function is a one-to-one

2b. Determine whether each function is a one-to-one

function.

function.

Verifying if two functions are Inverse of each other Suppose that f (x) is a one-to-one function. g(x) is the inverse of f(x) if 1.

for all x in the domain of g.

2.

for all x in the domain of f.

Or

where

is the inverse of

.

MDC Mathematics 2

EXAMPLE

YOUR TURN

Determine whether the two functions are inverses.

Determine whether the two functions are inverses.

3a. f ( x) = 2 x + 5 and g ( x) =

4a. f ( x) =

x −5 2

3 3 - 4 and g( x) = x x+4

x 3b. f ( x) = +2 and g ( x) = 3 x + 6 3

4b. f ( x) =

6 6 + 3 and g ( x) = x x +3

MDC Mathematics 3

The Process for Finding t he Inverse of a one to one Function Suppose that is a one-to one function, the equation of the inverse be found as follows: Step 1: Step 2: Step 3: Step 4:

can

Replace by y. Interchange x and y. Solve for y. Replace y by .

EXAMPLE

YOUR TURN

A one-to-one function is given. I. Find the inverse function. II. Find the Domain and the range of !!!"#!! !! III. Graph !!, ! !! !!"#!!! = ! on the same coordinate axes.

A one-to-one function is given. I. Find the inverse function. II. Find the Domain and the range of !!!"#!! !! III. Graph !!, ! !! !!"#!!! = ! on the same coordinate axes.

6a. f ( x) = 5 x − 4

6b. h( x) = −3x + 7

MDC Mathematics 4

EXAMPLE

YOUR TURN

A one-to-one function is given. I. II. III.

7a. f ( x) =

A one-to-one function is given.

Find the inverse function. Find the Domain and the range of !!!"#!! !! Graph !!, ! !! !!!"!!! = ! on the same coordinate axes.

1 x −7 2

I. II. III.

Find the inverse function. Find the Domain and the range of !!!"#!! !! Graph !!, ! !! !!"#!!! = ! on the same coordinate axes.

1 7b. g ( x) = 5 − x 3

MDC Mathematics 5

EXAMPLE

YOUR TURN

A one-to-one function is given. I. II. III.

A one-to-one function is given.

Find the inverse function. Find the Domain and the range of !!!"#!! !! Graph !!, ! !! !!"#!!! = ! on the same coordinate axes.

10a. f ( x) = x 2 + 4 for x ≥ 0

I. II. III.

Find the inverse function. Find the Domain and the range of !!!"#!! !! Graph !!, ! !! !!"#!!! = ! on the same coordinate axes.

10b. f (x) = x 2 − 1 for x ≥ 0

MDC Mathematics 6

EXAMPLE

A one-to-one function is given. I. Find the inverse function. II. Find the Domain and the range of !!!"#!! !! 9a. f ( x) =

3 − 1 x

YOUR TURN

A one-to-one function is given. I. Find the inverse function. II. Find the Domain and the range of !!!"#!! !! 9b. f ( x) =

5 + 2 x

EXAMPLE

A one-to-one function is given. I. Find the inverse function. II. Find the Domain and the range of !!!"#!! !! 11a. f ( x) =

x+5 x− 2

YOUR TURN

A one-to-one function is given. I. Find the inverse function. II. Find the Domain and the range of !!!"#!! !! 11b. h( x) =

2x −3 x +1

MDC Mathematics 7...