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Types of Functions Algebraic Functions MATH 1170 Chapter 1 Worksheet #1 NAME Note: It I bolded and underlined a term, you are responsible for a verbadom definition of that term (as well as understanding that definition). If I just bolded a term, I only expect that you are comfortable with the use of and using that term (i.e. understanding it). Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of fumctions. Section 1.2 of the text outlines a variety of types of functions. Notice that since the following are all functions, they will all pass the Vertical Line Test. Algebraic Functions A function is called an algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division and taking roots). Polynomials, power functions, and rational function are all algebraic functions. 1 Polynomials A function p is a polynomial if n n−1 2 p(x) = anx + an−1x + ::: + a2x + a1x + a0 where n is a nonnegative integer and a0; a1; a2; :::; an−1; an are all constants called coefficients of the polynomial. If the leading coefficient an 6= 0 then the degree of p(x) is n. • A polynomial of degree 1 is called a linear function. • A polynomial of degree 2 is called a quadratic function. • A polynomial of degree 3 is called a cubic function. • A polynomial of degree 4 is called a quartic function. • A polynomial of degree 5 is called a quintic function. • ...and so on... We often talk about finding roots of a polynomial. This means we are finding where the graph of the function hits the x-axis. Finding the roots of a polynomial entails setting the polynomial equal to zero and solving for x. Find the roots of p(x) = x2 − x − 6: 1 Linear Functions The most famous polynomial is the linear function. y is said to be a linear function of x if the graph of the function is a line so that we can use the slope-intercept form of the equation of a line to write a formula for the function as y = mx + b where m is the slope and b is the y-intercept. If this is the case, what do m and b equal in the p(x) equation? Graph the family of equations f(x) = x+b where be is an integer b = −2; −1; 0; 1; 2 on the same coordinate system. Graph the family of equations f(x) = mx where be is an integer m = −2; −1; 0; 1; 2 on the same coordinate system. Find the equation for a line that passes through (2; −1) and (3; 5). 2 Power Functions A function of the form f(x) = xa where a is a constant is called a power function. The power function takes a variety of forms based on the type of constant that a is. These different forms arrise when • 2.1 a = n where is n a positive integer, • 2.2 a = 1=n where n is a positive integer, and • 2.3 a = −1. We will explore these forms in the following sections. 2 2.1 If a = n where n is a positive integer... Notice that is n is a positive integer then the power function is really just a type of polynomial. Using your graphing calculator, sketch a graph of the following functions: 2 a. f1(x) = x b. f2(x) = x 3 4 c. f3(x) = x d. f4(x) = x 5 6 e. f5(x) = x f. f6(x) = x When n is odd, • what is the domain of f(x) = xn? • what is the range of f(x) = xn? • where is f(x) = xn increasing? • where is f(x) = xn decreasing? • How do you know that these characteristics will hold for every odd n? When n is even, • what is the domain of f(x) = xn? • what is the range of f(x) = xn? • where is f(x) = xn increasing? • where is f(x) = xn decreasing? • How do you know that these characteristics will hold for every even n? 3 2.2 If a = 1=n where n is a positive integer... The functions of the form f(x) = x1=n are called root functions. It is important to note that root functions are the inverses of polynomial functions. But what does this mean? Recall the definition of an inverse function from one of our previous lectures. (It can also be found on page 62 of the Stewart text.) What does it mean to say that root functions are the inverses of polynomial functions? Demonstrate that g(x) = x1=2 is the inverse function of f(x) = x2. Write f(x) = x1=n in a different form. Is f(x) = x1=n, where n is a positive integer, a polynomial? Using your graphing calculator as a tool, sketch a graph of the following functions: 1=2 1=3 a. f2(x) = x b. f3(x) = x 1=4 1=5 c. f4(x) = x d. f5(x) = x When n is odd, • what is the domain of f(x) = x1=n? • what is the range of f(x) = x1=n? • where is f(x) = x1=n increasing? • where is f(x) = x1=n decreasing? • How do you know that these characteristics will hold for every odd n? 4 When n is even, • what is the domain of f(x) = x1=n? • what is the range of f(x) = x1=n? • where is f(x) = x1=n increasing? • where is f(x) = x1=n decreasing? • How do you know that these characteristics will hold for every even n? 2.3 If a = −1... In this case, f(x) = x−1 is the reciprocal function. Is f(x) = x−1 a polynomial? Using your graphing calculator as a tool, sketch a graph of f(x) = x−1 and describe the domain, range and intervals of increasing and decreasing: Domain: Range: Increasing: Decreasing: 3 Rational Functions Moving on from power functions, we will now explore our last type of algebraic function: the rational function. A rational function is a ratio of two polynomials, p(x) and q(x): p(x) f(x) = q(x) Which of the previously mentioned functions is a rational function? What happens when you evaluate this function at x = 0? 5 x−1 Consider f(x) = x2−4 . • Identify p(x) and q(x). • Using your graphing calculator, sketch a graph of f(x). • What happens at x = 2 and x = −2? Why? Exponential Functions The exponential functions are the functions of the form f(x) = ax, where the base a is a positive constant. Note that these function are called exponential functions because the variable, x, is in the exponent. Using your graphing calculator as a tool, sketch a graph of the following functions and describe the domain, range and intervals of increasing and decreasing: a. f(x) = 2x b. f(x) = (1=2)x c. f(x) = 3x d. f(x) = (1=3)x When a > 1, • what is the domain of f(x) = ax? • what is the range of f(x) = ax? • where is f(x) = ax increasing? • where is f(x) = ax decreasing? • How do you know that these characteristics will hold for every a > 1? 6 When 0 < a < 1, • what is the domain of f(x) = ax? • what is the range of f(x) = ax? • where is f(x) = ax increasing? • where is f(x) = ax decreasing? • How do you know that these characteristics will hold for every 0 < a < 1? What is the difference between the function f(x) = x2 and g(x) = 2x? Find the exponential function f(x) = ax whose graph goes through the point (−4; 1=16): Logarithmic Functions The logarithmic functions, f(x) = logax, where the base a is a positive constant, are the functions that are the inverse of the exponential functions. It is important to realize that logarithmic functions are the inverses of exponential functions. x What does it mean to say that f(x) = logax is the inverse of g(x) = a ? x Show that f(x) = log2x is the inverse of g(x) = 2 .(Hint: This may require you to use the laws of logarithms that can be found on page 65 on Stewart.) 7 Using your graphing calculator as a tool, sketch a graph of the following functions and describe the domain, range and intervals of increasing and decreasing: a. f(x) = log2x b. f(x) = log10x Domain: Domain: Range: Range: Increasing: Increasing: Decreasing: Decreasing: Trigonometric Functions Using your graphing calculator as a tool, sketch a graph of the following functions and describe the domain, range and intervals of increasing and decreasing: a. f(x) = sinx b. f(x) = cosx c. f(x) = tanx Domain: Domain: Domain: Range: Range: Range: Increasing: Increasing: Increasing: Decreasing: Decreasing: Decreasing What do you notice about the graph of: • f(x) = sin(x + π=2)? • f(x) = sin(x + 2π)? 8 Piecewise Functions Piecewise functions are defined to be one of the above types of functions on one part of the x-axis and another function on a different part of the x-axis. For example consider ( x + 1; if x ≤ 1 f(x) = x2; if x > 1 This function has the same outputs as g(x) = x + 1 for x values less than of equal to 1 (the left half of the graph) and looks like h(x) = x2 for x greater than 1 (the right half of the graph). Sketch a graph of f(x). Find f(−2).
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