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##### Definition of Functions

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**Definition of Functions**The basic object of study in calculus is a function. A function is a rule or correspondence which associates to each number x in a set A a unique number f(x) in a set B. The set A is called the domain of f and the set of all f(x)'s is called the range of f.**Four Representations of a Function**• Symbolic or algebraic • Numerical • Graphical • Verbal**Continuity**If the graph of f breaks at x=x0, so that you have to lift the pencil off the paper before continuing, then f is said to be discontinuous at x=x0. If the graph doesn’t break at x=x0, then f is continuous at x0.**Increasing and Decreasing Functions**A function is increasing on an interval, if for any a, and b, in the interval f(a) < f(b) whenever a < b. In other words, If the curve of a line is going up from left to right, then it is increasing. A function is decreasing on an interval, if for any a, and b, in the interval f(a) > f(b) whenever a < b**One-to-one and Non-one-to-one Functions**A one-to-one function maps different inputs to different outputs. In other words, no two x values have the same y value. A non-one-to-one function can map more than one input to the same output.**Horizontal Line Test**A curve that passes the vertical line test, and thus is the graph of a function, will further be the graph of a one-to-one function if and only if no horizontal line intersects the curve more than once.**Basic Functions**Constant functions Power functions Inverses of functions Exponential functions Logarithm functions**Inverses of Functions**Let f be a function with domain D and range R. A function g with domain R and range D is an inverse function for f if, for all x in D, y = f(x) if and only if x = g(y).**Exponential Functions**Let b be a positive real number. An exponential function with base b is the function: f(x) = bx If the base of the function is e=2.7182818…, a particular irrational number (an infinite non-repeating decimal) between 2.71 and 2.72, the function ex is often referred to as the exponential function.**Logarithm Functions**Let b be a positive real number. The logarithm with base b f(x) = logb(x) is the inverse of the exponential function g(x) = bx. Informally, we can think of logb(x) as the exponent to which the base b must be raised to give x.**Natural Logarithm**The natural logarithm f(x) = ln(x) is the inverse of the exponential function g(x) = ex. In other words, ln(x) = loge(x).**Laws of Exponents and Logarithms(p. 26)**ln ex = x and elnx = x ex ey = ex+y ex/ey = ex-y e -x = 1/ex (ex)y = exy ln ab = ln a + ln b ln (a/b) = ln a – ln b ln (1/a) = - ln a ln (ab) = b ln a**Problem - Solving Tactic**To solve an equation with the unknown in an exponent, take the logarithm of both sides. To solve an equation with the unknown inside a logarithm, exponentiate both sides.**Solving Inequalities Involving Elementary Functions**Step 1: Find values of x where f is discontinuous. Step 2: Find the values of x where f is zero. Step 3: Look at the open intervals in between. On each of the intervals, f maintains only one sign.