MAC 1105 -2.7 Inverse Functions PDF

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Inverse Functions...

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2.7

ONE-TO-ONE FUNCTIONS AND INVERSE FUNCTIONS

Recall that functions have the special restrictions that each x value can only have one unique y value. We have an even MORE restrictive function called the ONE-TO-ONE FUNCTION. For a function to be a one-to-one function it must not only have the condition that each x has only one y (definition of a function) BUT it also has to have the restriction that EACH Y VALUE HAS ONLY ONE UNIQUE X VALUE. For a function, we could have such ordered pairs as: (2,3) and (-4,3) as long as the same x does not get two different y values. Now, for a one-to-one function, those points would not be allowed. Each variable must be in a one to one correspondence. No two x’s with the same y AND no two y’s with the same x.

If you are asked whether or not a function is one-to-one and you are given its equation, try to find two x values that produce the same y value. If you can’t, you have a one-to-one function.

EXAMPLES: 2x – 3y = -12 is one to one because each x value only yields one unique y value. However, y = x2 + 5 is NOT a one to one function because two different x values (like: x = -3 and x = 3) would produce the same y value, namely 14. So we would get ordered pairs: (-3, 14) and (3, 14).

If given a graph, it is easy to determine if it is one-to-one. We simply do a HORIZONTAL LINE TEST, very similar to the vertical line test for deciding if a graph is a function. If any horizontal line intersects the graph of a functions in no more than one point, then the function is one-to-one. If a horizontal line does cross over the graph more than once, it is NOT a one-to-one function.

1

INVERSE FUNCTIONS Next, we will consider

TWO

functions and decide if they are INVERSES of each other. We

are familiar with the idea of inverses in other parts of math. For instance, adding and subtracting are inverse operations of each other. Foiling and factoring are the inverse or “undoing” of each other. Similarly, we can sometimes have two functions that are inverses of each other. The functions must each be one-to-one in order to even try to be inverses of each other. REMEMBER, ONLY ONE-TO-ONE FUNCTIONS HAVE INVERSES. To decide if two functions are inverses, we have a test to perform. Functions f(x) and g(x) are INVERSES of each other IF:  f  g (x)  x AND  g  f (x)  x

EXAMPLES: Find f(g(x)) and g(f(x)) and determine whether the pair of functions f and g are inverses of each other. 1.

𝑓(𝑥) = 4𝑥 + 9

and

𝑔(𝑥) =

2.

𝑓(𝑥 ) = 3𝑥 − 7

and

𝑔(𝑥 ) =

𝑥− 9 4

𝑥+ 3 7

We denote an INVERSE function with special notation: f -1 (x) = “the inverse of function f”. Do not confuse this notation with a negative exponent.

f 1 ( x) 

1 f( x)

So if g(x) and f(x) are inverses of each other, then we can say that g(x) = f -1(x). 2

We must be able to take a one-to-one function and find its inverse. Here are the steps:

HOW TO FIND AN INVERSE OF A FUNCTION A.

Make sure the function is one-to-one first. (no squaring, abs value, etc.)

B.

Replace

C.

Exchange x for y and y for x. Everywhere you see an

f(x) or whatever functional notation is given with the letter y.

everywhere you see a y, change it to an D.

Solve for y.

E.

Replace

F.

Check that

-1

y with the f

x , change it to y and

x.

(x) notation.

f(x) and f

-1

(x) are inverses using the composition each way.

Remember the definition of inverses says:

f(

EXAMPLE:

f

-1

(x)

)

= x

f -1(

AND

Find the inverse of the function:

f(x)

)

= x.

f(x) = (x-2)3

A. Is it one-to-one? YES. Each value of x will only give one unique value of f(x). B. f(x) = y so make the function read:

y = (x-2)3

C. Interchange x for y. So this becomes

x = (y-2)3

D. Solve for y:

x  ( y  2) 3 cube root both sides 3

x 

3x

3

3

 y  23

 y 2

CHECK:

solve for y

x 2  y

3x

 2  f 1 (x) 3

EXAMPLES: These functions are one-to-one. For each, A) Find an equation for f -1(x), the inverse function. B) Verify your equation is correct by showing that f(f and f -1(f(x)) = x. 3.

f(x) = 3x - 1

4.

f(x) = ( x - 5) 3

5.

f(x) =

2𝑥 + 1 𝑥− 4

-1

(x)) = x

(IF TIME PERMITS)

4

Remember, that if a function is one – to – one, then its graph must pass the HORIZONTAL LINE TEST. Also, only one – to – one functions have inverses. Therefore, we can use the same Horizontal Line Test to decide if a graph of a function has an inverse function. 6. Does each graph represent functions that have inverse functions? Why or Why not?

When you graph a function and its inverse on the same set of axes, you see a very pretty symmetry they have around the line x = y. Do you see why this would be? (Hint: Look at step #3 on the list of how to get an inverse). Also, it is important to note that when you have two functions that are inverses of each other, the DOMAIN of one is the RANGE of the other AND the RANGE of the one is the DOMAIN of its inverse. This is again because x became y and y became x. 7.

Graph the inverse of the function shown below. Label at least three points on the graph of the inverse.

5

8.

Graph the inverse of the function shown below. Label at least three points on the graph of the inverse.

For the function:

f(x) = 2x - 3:

9.

EXAMPLE:

A)

Find an equation for f

B)

Graph f(x) and f

C)

Use interval notation to give the domain and range of both.

-1

-1

(x).

(x).

6

10.

EXAMPLE:

Let f(x) and g(x) be defined by the following tables. Use the tables to evaluate each composite function.

x

f(x)

x

g(x)

-1 0 1 2

1 4 5 -1

-1 1 4 10

0 1 2 -1

A)

f( g(4) )

11.

EXAMPLE:

B)

Let:

( 𝒈 ° 𝒇)(𝟎)

f(x) = 2x – 5

g(x) = 4x - 1

and

Find: A)

( 𝒈 ° 𝒇)(𝟑)

B)

( 𝒇 ° 𝒈)(𝟑)

7...