Lecture 5: Composition of functions

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Assignment

1.6        Inverse Functions and   1-9, 19-21, 27-30,      1/30              
           Logarithms              48-52                                     

1.7        Models and Curve        1                       1/30              
           Fitting                                                           

Chapter 1  review problems         1-4,7-8, 10-11, pages   1/30              
                                   92-3                                      

Suppose f(x+2)= 3x2+12x +12. Can you find f(x)? Students found several ways to solve this in class. Here's another way. Notice that f(x+2) is the composition of the two functions f and g, where g(x)=x+2. Since g has an inverse function (what is it?), we can compute fogog-1 and just get f . OK, g-1(x)=x-2, so fogog-1(x)=fog(x-2)=f(x-2+2)=3(x-2)2+12(x-2)+12.

Try the same method with f(x+2)=10x. IE, find f(x).

Brief review of the lecture (left from a math 1100 lecture)

• One of the most important types of algebraic expressions is called a polynomial. The term polynomial (of a single variable) was defined here as a sum of multiples of powers of a variable.
• Polynomials may be classified according to
  1. Number of variables (for our purposes, this will almost always be 1)
  2. Degree, e.g., the highest power of all the terms of the polynomial.

1. Example : x² - xy²z+10 has 3 variables and degree 4

• Multiplication of polynomials was discussed, both from the algebraic and geometric viewpoints.

1. Example : (x-2)(x²-3x+5)=x³-5x²+11x-10

• Decimals were defined as a sum of multiples of powers of ten.

1. Example : 401.03= 4 × 10 ² + 1 × 10 ° + 3 × 10 ¯ ²

• Polynomials were defined as a sum of multiples of powers of a variable.

1. Example : 4x²+3x+9

• The following table shows how polynomials are ofter classified by their degree:

• Degree	Symbolic representation	Common representation	Name	Maximum # of zeros
  0	a0	c	constant	none
  1	a1x+a0	mx+b	linear	1
  2	a2x2+a1x+a0	ax2+bx+c	quadratic	2
  3	a3x3+a2x2+a1x+a0	none	cubic	3
  4	as expected	none	quartic	4

  
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