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Function Compositions and Inverses

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Function Compositions and Inverses

- f(x) does NOT stand for MULTIPLICATION!
- We read f(x) “f of x” and it means that the function’s independent variable is x.
- If the function is defined as f(x) = 3x - 1and we are asked to find f(2), we just substitute2 for x in the function:

f(2) = 3(2) - 1 = 6 - 1 = 5

Given g(x) = x2 - x, find g(-3)

Substitute -3 for x:

g(-3) = (-3)2 - (-3)

= 12

g(-3) = 12

Given g(x) = 3x - 4x2 + 2, find g(5)

Substitute 5 for x

g(5) = 3(5) - 4(5)2 + 2

= -83

g(5) = -83

Function Composition is just more substitution, very similar to what we have been doing with finding the value of a function. The difference is we will be substituting another function instead of a number ...

For example…

Given f(x) = x - 5, find f(a+1)

Substitute (a+1) for x

f(a + 1) = (a + 1) - 5

= a+1 - 5

= a - 4

The answer is a function in terms of ‘a’

- Composition notation looks like g(f(x)) or f(g(x)), we read this ‘g of f of x’ or ‘f of g of x’.
- We are given f(x) and g(x), the function inside the parentheses gets substituted into the other.

Given the functions:

f(x) = 2x+2 & g(x) = 2

find f(g(x))

This notation tells us to substitute the g(x) function, 2, for x in the f(x) function:

f (2) = 2(2)+2

= 6

Given the functions:

g(x) = x - 5 & f(x) = x + 1

find f(g(x))

This notation tells us to substitute the g(x) function, x-5, for x in the f(x) function:

f (x-5) = (x-5)+1

= x - 4

Reverse the composition:

g(x) = x - 5 & f(x) = x + 1

find g(f(x))

This notation tells us to substitute the f(x) function, x+1, for x in the g(x) function:

g(x+1) = (x+1)-5

= x - 4

In the last example, f(g(x)) and g(f(x)) had the same results. This is not always the case.

Try this example:

f(x) = x2 + x & g(x) = x - 4

find f(g(x)) and g(f(x))

f(x) = x2 + x & g(x) = x - 4

1) f(g(x)) = f(x-4) = (x-4)2 + (x-4)

= x2 -8x+16+x-4

= x2 -7x+12

2) g(f(x)) = g(x2 + x ) = (x2 + x )-4

= x2 + x - 4

New Example:

Given f(x) = 2x + 5 & g(x) = 8 + x

find f(g(-5) & g(f(-5)

1) f(g(-5) : find g(-5) = 8 + (-5) = 3

then find f(3) = 2(3) + 5 = 11

2) g(f(-5)) : find f(-5) = 2(-5) + 5 = -5

then find g(-5) = 8 + (-5) = 3

Remember: a function is a set of ordered pairs (including lists of discrete points and also equations which give us infinite points), where no two points have the same x-coordinate.

The Inverse of a function is the set of points where each point in the function is reversed, (y, x).

A function that is a list of ordered pairs is easy to find the inverse of:

f(x) = {(1, 2), (2, 5), (3, -4), (4, 0)}

The inverse is:

f-1(x) = {(2, 1), (5, 2), (-4, 3), (0, 4)}

To find the inverse of a function that is written as an equation, like:

f(x) = x + 7

We will:

1) Replace the function label, f(x) with y

2) Swap the variables, x and y

3) Solve the new equation for y

Find the Inverse of:

f(x) = x + 7

Replace: y = x + 7

Swap: x = y + 7

Solve:y = x - 7

f-1(x) = x - 7

Find the Inverse of:

f(x) = 3x - 4

Replace: y = 3x - 4

Swap: x = 3y - 4

Solve:3y = x + 4 y = (x + 4)/3

f-1(x) = (x + 4)/3

Find the Inverse of:

f(x) = (2x + 5)/3

Replace: y = (2x + 5)/3

Swap: x = (2y + 5)/3

Solve:3x = 2y + 5 2y = 3x - 5

f-1(x) = (3x - 5)/2

Find the Inverse of:

f(x) = x2 - 4

Replace: y = x2 - 4

Swap: x = y2 - 4

Solve:y2 = x + 4 y = ±√x + 4

Note: In the last example, the inverse does NOT pass the test to be a function. This sometimes happens, that the inverse of a function is not a function.

This occurs when the function has points with the same y-value (allowed in functions).

A function whose inverse IS ALSO a function is called a ONE-TO-ONE function.

Each x-coordinate has a different y-coordinate and each y-coordinate has a different x-coordinate.