1.3: The Basic Classes of Functions

This section outlines the different types of important functions in calculus. There are polynomials, which have whole number exponents and consist of a bunch of numbers added together. There are also rational functions, defined by P(x)/Q(x) where Q(x) is not equal to zero, because it would make the function undefined. Algebraic functions involve a variety of operations, polynomials and rational functions. They are kind of like a combination of the other types of functions outlined in this section. Exponential functions are functions that are always positive and involve bases raised to a specific exponent. They are the inverse of Logs, or logarithmic functions, which can be used to find an unknown exponent. Both exponential and logarithmic functions are found in everyday life, some examples being the Richter scale, pH levels, the economy, C13 dating, etc. Section 1.3 also describes how to construct new functions. New functions can be formed by adding functions, subtracting them, multiplying and dividing them. Composition of functions occurs when you have a value for f(x) and a value for g(x) and you combine them by plugging the value of g(x) in for the value of x in f(x). For example if f(x) = x^2 and g(x) = 5-x and you want to find out what f(g(x)) is equal to; you would write (5-x)^2 and solve for x. This can also be turned around to find g(f(x)).
Square root of 1-t ^3

Problems:

#34: Calculate the composite functions f(g(x)) and g(f(x)) and determine their domains.

F(t) = √t g(x) = 1-t^3

F(g(x)= 〖√(1-t)〗^3D: DNE for any value of t. g(f(x))=1-√t ^3 D: (0, infinity)

To solve this problem, I started with f(g(x)). I plugged the value of g(x) in for t in the function f(x). To solve g(f(x)) I did the same thing, but opposite, I plugged the value of f(x) in for t in the function g(x). To determine the domains of the functions I tried various numbers for t and carried out the function. I found that every value I tried for f(g(x)) left me with a non real answer, allowing me to come to the conclusion that there was no possible value for t which would make this function work. For the function g(f(x)) I did the same thing, using random values for t. I found that negative numbers would not work, because you cannot take the square root of a negative number, so I concluded that the domain would have to be from zero to infinity.

#36: Find all values of c such that the domain of f(x)= (x+1)/(x^2+2cx+4) is all real numbers.

I think that all real numbers will work when plugged in for c to keep the domain of the function all real numbers. I plugged a variety of numbers in for c, using x =2 and every value that I found fit into the category all real numbers.