Chapter 1 Functions and Models 1.1 Four Ways to Represent a Function Definition. A function f is a rule that assigns to each element x in a set D exactly one element, called f(x), in a set E. • The set D is called the domain of the function, i.e., the set of all possible x's. • The range of f is the set of all possible values of f(x) as x varies throughout the domain. f(a + h) − f(a) Example 1.1. If f(x) = x2 + 3x + 2 and h =6 0, evaluate . h 1 2 CHAPTER 1. FUNCTIONS AND MODELS Example 1.2. A rectangle has area 16 m2. Express the perimeter of the rectangle as a function of the length of one of its sides. 1.1.1 Piecewise Defined Functions Example 1.3. A function f is defined by 8 < 1 3 − x if x ≤ 2 f(x) = : 2 2x − 5 if x > 2 Evaluate f(−2), f(0), and f(2) and sketch the graph. f(x) 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 5 −1 −2 −3 −4 −5 1.1. FOUR WAYS TO REPRESENT A FUNCTION 3 1.1.2 Symmetry • If a function f satisfies f(−x) = f(x) for every number x in its domain, then f is called an even function. • If f satisfies f(−x) = −f(x) for every number x in its domain, then f is called an odd function. Example 1.4. Determine whether each of the following functions is even, odd, or neither odd nor even. a) f(x) = x5 + x b) g(x) = 1 − x4 c) h(x) = 2x − x2 4 CHAPTER 1. FUNCTIONS AND MODELS 1.1.3 Domain Example 1.5. Find the domain of each function. 2 − − 2 7 a) F (x) = x 2x + 1 b) H(t) = 4 t c) g(t) = p 2 − t 1.2. MATHEMATICAL MODELS 5 1.2 Mathematical Models A mathematical model is a mathematical description- often by means of a function or an equation- of a real-world phenomenon such as size of a population, demand for a product, speed of a falling object, life expectancy of a person at birth, or cost of emission reductions, etc. There are many different types of functions that can be used to model relationships observed in the real world. 1.2.1 Linear Models When we say that y is a linear function of x, we mean that the graph of the function is a line. So, we can use the slope-intercept form of the equation of a line to write a formula for the function as y = f(x) = mx + b; where m is the slope of the line and b is the y-intercept. A characteristic feature of linear functions is that they grow at a constant rate. Example 1.6. The manager of a furniture factory finds that it costs $2200 to manufacture 100 chairs in one day and $4800 to produce 300 chairs in one day. a) Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch a graph. 18 16 14 12 10 8 6 b) What is the slope of the graph and what does it represent? ), in thousands 4 n ( C 2 0 0 50 100 150 200 250 300 350 400 450 n, chairs c) What is the y-intercept and what does it represent? 6 CHAPTER 1. FUNCTIONS AND MODELS 1.2.2 Power Functions A function of the form f(x) = xa, where a is constant, is called a power function. Example 1.7. Sketch the graphs of each power function using your calculator or library of functions. a) y = x b) y = x2 c) y = x3 d) y = x4 e) y = x5 y y y 3 3 3 2 2 2 1 1 1 x x x −3 −2 −1 1 2 3 −3 −2 −1 1 2 3 −3 −2 −1 1 2 3 −1 −1 −1 −2 −2 −2 −3 −3 −3 Example 1.8. Sketch the graphs of each power function using your calculator or library of functions. p p 3 1 a) f(x) = x b) g(x) = x c) h(x) = x y y y 3 3 3 2 2 2 1 1 1 x x x −3 −2 −1 1 2 3 −3 −2 −1 1 2 3 −3 −2 −1 1 2 3 −1 −1 −1 −2 −2 −2 −3 −3 −3 1.2. MATHEMATICAL MODELS 7 1.2.3 Polynomials A function P is called a polynomial if n n−1 P (x) = anx + an−1x + ··· + a1x + a0; where n is a nonnegative integer and the numbers a0; a1; a2; : : : ; an are constants called the coefficients of the polynomial. A polynomial is a sum or difference of power functions when n ≥ 0. 1.2.4 Rational Functions A rational function f is a ratio of two polyno- mials P (x) ; Q(x) where P and Q are polynomials, and Q =6 0. 2x4 − x2 + 1 Example 1.9. For the function f(x) = , graph on your graphing x2 − 4 calculator, and sketch it on the grid. 1.2.5 Algebraic Functions A function f is called an algebraic function if it can be constructed using algebraic operations (addition, subtraction, multiplication, division, and taking roots) starting with polynomials. 8 CHAPTER 1. FUNCTIONS AND MODELS p p Example 1.10. Graph y = x x + 3, y = 4 x2 − 25, and y = x(2=3)(x − 2)2 on your calculator and sketch it on the grids. 1.2.6 Trigonometric Functions In calculus, the convention is that radian measure is always used (except when otherwise indicated). f(x) = sin x and g(x) = cos x • For both the sine and cosine functions, the domain is (−∞; 1) and the range is the closed interval [−1; 1]. • For all values of x we have −1 ≤ sin x ≤ 1 and −1 ≤ cos x ≤ 1 • The sine and cosine functions are periodic functions and have a period 2π. • The zeros of the sine functions occurs when x = nπ. • The tangent function is related to the sine and cosine functions by the equation sin x tan x = . cos x • The remaining three trigonometric functions: cosecant, secant, and cotangent are the reciprocals of the sine, cosine, and tangent functions. 1.2. MATHEMATICAL MODELS 9 1.2.7 Exponential Functions The exponential functions are the functions of the form f(x) = ax, where the base a is a positive constant. Example 1.11. Graph y = 2x and y = 0:5x on your calculator and sketch it on the grid. 1.2.8 Logarithmic Functions The logarithmic functions are the functions of the form f(x) = loga x, where the base a is a positive constant. The logarithmic functions are the inverse functions of the exponential functions. Example 1.12. Graph y = log x and y = 10x on your calculator and sketch it on the grid. How do the two functions relate? 10 CHAPTER 1. FUNCTIONS AND MODELS 1.3 New Functions from Old Function 1.3.1 Transformations of Functions Recall. A general function with transformations looks like af(x − h) + k. Let f(x) be a function. Suppose c > 0. To obtain the graph of f(x) c f(x c) Move c units to up/down, Move c units to the left/right, add/subtract c units to each add/subtract c units from each x- y-coordinate, and f(x) has a coordinate, and f(x) has a hori- vertical shift zontal shift p Examplep 1.13. Given the graphpf(x) = 4 − x2, use transformations to graph g(x) = 4 − x2 − 3 and h(x) = 4 − (x − 1)2. g(x) h(x) 5 5 4 4 3 3 2 2 1 1 x x −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 −1 −1 −2 −2 −3 −3 −4 −4 −5 −5 1.3. NEW FUNCTIONS FROM OLD FUNCTION 11 Let f(x) be a function. Suppose c > 1. To obtain the graph of cf(x) f(cx) −f(x) f(−x) Stretch/shrink Stretch/shrink Reflection about Reflection about f(x) vertically by a f(x) horizontally the x-axis, multiply the y-axis, multiply factor of c, multiply by a factor of c, di- y-coordinates by a x-coordinates by a y-coordinates by c vide x-coordinates negative negative by c p Examplep 1.14. Givenp the graph f(x) = p4 − x2, use transformations to graph g(x) = − 4 − x2, h(x) = 2 4 − x2, and k(x) = 4 − (−x)2. g(x) h(x) k(x) 4 4 4 3 3 3 2 2 2 1 1 1 x x x −4 −3 −2 −1 1 2 3 4 −4 −3 −2 −1 1 2 3 4 −4 −3 −2 −1 1 2 3 4 −1 −1 −1 −2 −2 −2 −3 −3 −3 −4 −4 −4 1 Example 1.15. Sketch the graph f(x) = cos x and g(x) = 2 − cos x. 2 f(x) g(x) 3 3 2 2 1 1 x x −4π −3π −2π −π π 2π 3π 4π − −3π − −π π 3π −1 2π 2 π 2 −1 2 π 2 2π −2 −2 −3 −3 12 CHAPTER 1. FUNCTIONS AND MODELS 1.3.2 Combinations of Functions We say (f ◦ g)(x) = f(g(x)) as the function f is composed with g, i.e, we substitute every x in f with the function g(x).