Composite Functions
Yesterday we combined functions in several different ways:
Example If f(x) = x2 and g(x) = x + 3, determine:
a) f(x) + g(x) b) f(x) - g(x) c) f(x)g(x)
Composing functions is simply another method of combining two functions.
Composite Functions
f(g(x)) or is the composite function that is formed when g(x) is substituted for x in f(x)
f(g(x)) is pronounced "f of g of x"
Note: g(f(x)) is the composite function that is formed when f(x) is substituted for x in g(x)
Ex. 1 If f(x) = 2x + 3 and g(x) = , determine:
a) f(g(9)) b) g(f(23.5)) c) f(g(x)) d)
Ex. 2 If f(x) = x2 and g(x) = x + 3,
a) determine the equation for the composite functions f(g(x)), g(f(x)), and g(g(x))
b) graph each composite function
c) state the domain and range of each composite function
Ex. 3 Let u(x) = x2 + 3x + 2 and w(x) = evaluate u(w(2)).
Ex. 4 Given that h(x) = f(g(x)) = :
a) If f(x) = what is g(x)? b) If g(x) = x2, what is f(x)?
Date: ______
Inverse Functions
The inverse of a function can be found by:
Notation: the inverse of the function f(x) is written f -1(x)
Ex. 1 Find the inverse of each function:
a) f(x) = 2x + 4
b)
c)
d)
What do you notice about the domain and range of each function compared to the domain and range of the inverse?
Is the inverse of a function always a function?
Ex. 2 Find the inverse of f(x) = x2 - 3
a) algebraically
b) graphically
c) Restrict the domain of the function so that the inverse is also a function.
Dividing Polynomials
Dividing a polynomial by a monomial is quite simple:
Ex.
But, how would we divide something like:
To find the quotient of this division we can use Polynomial Division.
Polynomial Division is a process similar to long division
Recall the steps in dividing:
Every division statement that involves numbers can be re-written as a multiplication and an addition.
Ex.
Let’s try a similar strategy to divide
*Note – You must divide 2 terms into 2 terms. But only divide the first term into the first, and then follow division rules.
Step 1: Divide the first term of the dividend by the first term of the divisor ---à by x ()
Step 2: Multiply the divisor by
Step 3: Subtract that product
Step 4: Carry down the next term (remember the note above)
Step 5: Repeat steps 1 – 4.
Once you have finished the division, write the multiplication statement that shows the divisor, dividend, quotient and remainder.
Dividend: D(x) =
Divisor: d(x) =
Quotient: Q(x) =
Remainder: R(x) = 14
Recall: D(x) = d(x) ∙ Q(x) + R(x)
= () + 14
Ex. 1 Divide the following
a)
*Note – You MUST fill in a into the dividend, or this won’t work!! Also, you must divide 3 terms into 3 terms, and again, the leading terms of the polynomials are what get divided.
b)
Note: If a polynomial is a factor of another polynomial, the remainder will be zero.
Synthetic Division
As a short cut when dividing a polynomial by a binomial in the form x - k (where k is the value of x that makes the divisor equal to 0) we can use a process called synthetic division.
Synthetic Division is a fast and efficient way to divide if the divisor meets the criteria mentioned above, but it will not work for ALL polynomial divisions.
Divide x3 - 3x2 + 4x - 8 by x - 2
i) using Long Division ii) using Synthetic Division
Ex. 1 Use synthetic division to divide:
a) (x4 - x2 + 2x - 3) ÷ (x + 2)
b) ( - 5x2 -10 + 6x +2x3 ) ÷ (2x - 1) c) (x3 - 6x2 + 12x - 8) ÷ (x - 2)
*Write the remainder as a fraction *Writ e the quotient in factored form
Remainder Theorem
Given f(x) = x3 - 3x2 - 2x + 5,
i) divide f(x) by x - 4 ii) evaluate f (4)
Given P(x) = 2x4 - 4x2 - 7x + 3,
i) divide P(x) by x +3 ii) evaluate P (-3)
What do you notice about the remainder and the value of the function at k? (Recall: k is the value of x that makes the divisor equal to 0)
The remainder theorem states that when a polynomial function P(x) is divided by a binomial (x – b), the remainder is P(b).
Furthermore, if a polynomial is divided by a binomial of the form (ax – b), the remainder is where a, b are integers and a0.
Ex. 1 Divide by (2x + 1) and then try
Ex. 2 Use the remainder theorem to find the remainder when is divided by:
a. (x + 1) b. (2x – 3)
Ex. 3 When x4 - 2x2 + kx - 5 is divided by x - 3, the remainder is 11. Find k.
Factor Theorem
Recall: Remainder Theorem When a polynomial P(x) is divided by x - b, the remainder is f (b).
We also know, that in order for a polynomial to be a factor of another polynomial, ______
______
The factor theorem states that (x – b) is a factor of a polynomial P(x) if and only if P(b) = 0.
The factor theorem allows you to determine the factors of a polynomial without having to divide.
Ex.1 Determine if (x – 3) and (x + 2) are factors of
Ex.2 a) Which binomials are factors of the polynomial
i. x – 2 ii. x + 2 iii. x + 1 iv. x – 1 v. 2x + 1
b) Use the results from part a) to write P(x) in factored form.
The factor theorem is VERY useful in factoring polynomials.
Once a factor is found, polynomial division can be used to find the other factor(s).
Finding the first factor is somewhat trial and error. Although, if the leading coefficient of P(x) = 1, then b must be a factor of the constant term of the polynomial.
Ex 3. Factor fully:
* Before anything you must name your polynomial
* We must find a value of x such that P(x) = 0
* The values (b) must be a factor of the constant term
* Now we must use trial and error to test the possible values of b to find a factor.
TEST
x P(x) Conclusion
Now, to find the other factor(s) divide the polynomial by the known factor:
Ex. 4 Factor the following fully:
Factor Theorem Continued
The factor theorem states that (x – b) is a factor of a polynomial P(x) if and only if P(b) = 0.
Similarly, (ax – b) is a factor of P(x) if and only if .
Recall: The factor theorem allows you to determine one factor of a polynomial and division can be used to find the other factor(s).
If the leading coefficient of P(x) ≠ 1, then you are looking for a rational value of to find P(x) = 0.
Here are the rules we use to help find the value of .
·
·
·
· b is a factor of the constant term of P(x)
· a is a factor of the leading coefficient of P(x)
· ax – b is a factor of P(x) if
·
Ex. Factor ***Note: Leading coefficient is 3,
the constant is 2.
· Possible values of b are ______
· Possible values of a are ______
· Possible values of are ______
***HINT*** Always test integer values before fractions
TEST
x V(x) Conclusion
Ex. 1 Determine the values that could be zeroes for the polynomial, then factor.
Factoring Sum and Difference of Cubes
Recall from previous grades that:
a. b. c.
This is the rule for factoring a difference of squares.
You also learned that a sum of squares () does not factor at all.
Ex. Factor the following
**Note if then
Now, let’s try to use polynomial division to see if we can factor a difference of cubes
We will try
**Note – We must include and because during the division, they will we will need to subtract from each of these.
**Note - Since the remainder is 0, we know that (x – y) is a factor, and we have successfully factored
Therefore, any difference of cubes can be factored using the following:
(It is always a good idea to figure out what A and B are before you factor - )
Although you cannot factor a sum of squares, a sum of cubes can be factored by using the same methodology as the difference of cubes.
Let’s try
Therefore, any sum of cubes can be factored using the following:
You absolutely MUST memorize the sum and difference of cubes formulae and be able to recognize factoring questions with sums and differences of cubes.
Ex. 1 Factor Fully
a. b.
c.
**Note – it may be helpful for you to memorize some of the perfect cube numbers to help you recognize them in questions. In order, the first 10 perfect cubes are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000. You will see these numbers OVER AND OVER.