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
- Number of variables (for our purposes, this will almost always
be 1)
- Degree, e.g., the highest power of all the terms of the polynomial.
- Example : x² - xy²z+10 has 3 variables and degree
4
- Multiplication of polynomials was discussed, both from the
algebraic and geometric viewpoints.
- Example : (x-2)(x²-3x+5)=x³-5x²+11x-10
- Decimals were defined as a sum of multiples of powers of ten.
- Example : 401.03= 4 × 10 ² + 1 × 10 °
+ 3 × 10 ¯ ²
- Polynomials were defined as a sum of multiples of powers of
a variable.
- 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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