Prep for Calculus, Sec 1.1: Basic Functions PDF

MATH 1813, L. Brown Prep for Calculus, Sec. 1.1: The Basics of Functions Informally, we say a function describes how one quantity depends on another quantity. There are multiple ways to represent a function: in words, by a formula, using a graph, or in the form of a table. Verbally...

MATH 1813, L. Brown Prep for Calculus, Sec. 1.1: The Basics of Functions Informally, we say a function describes how one quantity depends on another quantity. There are multiple ways to represent a function: in words, by a formula, using a graph, or in the form of a table. Verbally (Words): You deposit $1000 in a bank account that pays 3% annual interest. Let A be the amount of money you have in the account in t years. Algebraic (Formula): A = f (t ) = 1000(1.03)t , where A is the amount of money (in dollars) you have in the account in t years. Graphical: Numerical (Table): A dollars t A years dollars 0 1000.00 1 1030.00 f(2) = 1060 2 1060.00 f(2) = 1060 3 1092.73 4 1125.51 t years Function Definition: A function is a rule that assigns each value in one set (called the domain) exactly one value in another set (called the range). Example 1 (Video) Determine whether the formula describes y as a function of x. (Note: Assume x is the input value and y is the output value.) Explain your reasoning. x2 + y 2 = 1 1  1.1 The Basics of Functions Determining Whether a Graph Represents a Function If we look at the graph of x + y = 1 , we see it is a circle 2 2 of radius one centered at the origin. What x-values did you use in Example 1, and which of them helped you draw your conclusion? Vertical Line Test: If every vertical line intersects a graph at most once, then the graph represents a function. Your Turn 1 Determine whether the given graph represents a function y y y 3 3 3 2 2 2 1 1 1 x x x -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -1 -1 -1 -2 -2 -2 -3 -3 -3 If y is a function of x, then the notation we use is y = f(x). In this case, x is called the independent variable (input value) and y is called the dependent variable (output value). Example 2 (Video) If ℎ (𝑥 ) = 5(𝑥 − 2)2 + 1, then find each of the following. (a) ℎ(3) (b) ℎ(3𝑎) (c) ℎ(3 − 𝑎) Your Turn 2 (Video) If 𝑓(𝑥 ) = 3𝑥 2 + 𝑥 − 1, then find each of the following. (a) 𝑓(−2) (b) 𝑓(2𝑎) (c) 𝑓(𝑎 + ℎ) Example 3 (Video) Using the functions f, g, and h below, find each of the following. ℎ(𝑥 ) = −4𝑥 + 1 x –2 –1 0 1 2 f(x) 3 0 1 –2 –1 Graph of g (a) f(–2) = ______ (b) g(–2) = ______ (c) h(–2) = ______ (d) Solve f(x) = –2 for x. x = _________ (e) Solve h(x) = –2 for x. x = _________ (f) Solve g(x) = –2 for x. x = _________ 3  1.1 The Basics of Functions Piecewise Defined Functions Piecewise defined functions consist of different formulas for different parts of the domain. Example 4 (Video) For the piecewise defined function, find f(–6), f(–1), and f(3). 3 − 𝑥, 𝑥 < −1 𝑓 (𝑥 ) = { 2𝑥, 𝑥 ≥ −1 f(–6) = ________ f(–1) = ________ f(3) = ________ Your Turn 4 (Video) For the piecewise function g, find g(–5), g(–2), and g(8). −𝑥, 𝑥 < −2 𝑔(𝑥 ) = {1, −2≤𝑥