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Algebra 2 Notes
Ch. 8 – Inverses & Radicals
p. 1 of 6
8.1 – COMPOSITION OF FUNCTIONS
If f(x) = x2 – 2x + 3, then f(5) means putting 5 in the place of x.
So, f(5) = 52 – 2(5) + 3, or f(5) = ________.
What is f(x+2)? _____________.
How about f(3n)? ____________.
Composition of Functions
By f(g(x)), or the COMPOSITION of f and g, we mean the following:
1. Evaluate g(x)
2. Evaluate f of whatever the value of g(x).
In other words: f(the answer to g(x)).
Suppose f(x) = x + 5, and g(x) = x2.
To find f(g(3)),
1. Find g(3). g(3) = 32 or 9.
2. Find f(9). f(9) = ________. So f(g(3)) = _______.
** Just as with any ( ), you work from the inside out**
In order…given x, find g(x), use this to find f(g(x))
Find the following:
1. f(g(-1)) =
Algebra 2 Notes
Ch. 8 – Inverses & Radicals
p. 2 of 6
f(x) = x + 5
g(x) = x2
2. f(g(4)) =
3. f(g(8)) =
4. g(f(8)) =
5. g(f(x)) =
Alternate form: f(g(x)) can be written as (fg)(x). It still means do g(x) first.
Algebra 2 Notes
Ch. 8 – Inverses & Radicals
p. 3 of 6
8.2 INVERSES OF RELATIONS
I. What is an inverse?
Very generally, an inverse “undoes” or “gets you back to where you started.”
Ex.
The inverse operation for addition is _______________.
The inverse operation for multiplication is _______________.
The inverse operation for a 3rd power is _________________.
II. How do you find the inverse of a function?
A. If you know the ordered pairs which make up the function, to find the
inverse: _________________________________________.
Example:
f: {(3,7) , (4,2) , (5,-3) , (6,-9)}
Inverse of f: ____________________________.
B. If you know the graph of a function, to find the graph of the inverse,
___________________________________________________.
Examples:
C. If you know the equation of a function, to find its inverse:
___________________________________________________.
Example: f(x) = Y = 3x2 – 2x + 5
k(x) = Y = 2x – 3
(more on this in the 8.3 notes)
Inverse of f: __________________
Inverse of k: __________________
Algebra 2 Notes
Ch. 8 – Inverses & Radicals
p. 4 of 6
III. Under what conditions will the inverse of a function also be a function?
A. The function must be _______. This means that each ___ must be paired
up with exactly one ___ (and therefore it is a __________), and each ___
must be paired up with exactly one ___ (that’s the 1-1 part).
Ex.
{(2,4) , (5,7) , (1,-3) , (7,4)}
Function _____; 1-1 _____
{(2,4) , (5,7) , (2,-3) , (7,-4)}
Function _____; 1-1 _____
{(2,4) , (5,7) , (1,-3) , (7,-4)}
Function _____; 1-1 _____
B. If the graph is given, apply both the Vertical and the Horizontal Line Tests.
The Vertical Line Test to show the original is a function. The Horizontal
Line Test is applied to the __________ to show the __________ is a
function.
Examples:
Algebra 2 Notes
Ch. 8 – Inverses & Radicals
p. 5 of 6
8.3 – INVERSES OF RELATIONS
I. Notation
f
–1
is called the “inverse of the function f” and it is read “f inverse.”
Steps to finding an Inverse:
1. Substitute y for f(x)
2. Exchange the y and x
3. Solve back for y
4. Substitute f –1 (x) for y.
Given: f(x) = 3x + 5
Find f –1(x)
Given: f(x) = x – 9
4
Find f –1(x)
Given: f(x) = 5x7 + 4
Find f –1(x)
Algebra 2 Notes
Ch. 8 – Inverses & Radicals
p. 6 of 6
II. Working with inverses
Given: f(x) = 3x + 5
and
f –1(x) = x – 5
3
Find f(f –1(3)) =
Find f –1(f(5)) =
Find f –1(f(x)) =
Find f(f –1(x)) =
If f and g are functions such
that f(g(x)) = ___ and g(f(x)) =
___ for all x, then f and g are
inverse functions