Section 1.2 Combining Functions; Shifting and Scaling Graphs We will get new functions from the ones we know. Tow functions f and g can be combined to form new functions by function addition, substraction, multiplication, division and composition. (a) Function addition: Given two functions f and g we define the sum of f and g as (f + g)(x) = f(x) + g(x) Let A be the domain of f, B be the domain of g then domain of f+g is A B. (b) Function Substraction: Given two functions f and g we define the difference of f and g as (f g)(x) = f(x) g(x) Let A be the domain of f, B be the domain of g then domain of f-g is A B. Example 1 (a) Let f(x) = x 3 and g(x) = x 2. Find (f + g)(x), (f g)(x) and state their domains. (b) Let f(x) = x 2 3 and g(x) = 3 x 2. Find (f + g)(x), (f g)(x) and state their domains. Solution: (a) (f + g)(x) = x 3 + x 2, (f g)(x) = x 3 x 2. Domain for f + g and f g are the same and is [3, ). (b) (f + g)(x) = x 2 3 + 3 x 2, (f g)(x) = x 2 3 3 x 2. Domain for f + g and f g are the same and is { 3, 3}. (c) Function Multiplication: Given two functions f and g we define the multiplication of f and g as (fg)(x) = f(x)g(x) Let A be the domain of f, B be the domain of g then domain of fg is A B. (d) Function Division: Given two functions f and g we define the division of f and g as ( f f(x) )(x) = g g(x) Let A be the domain of f, B be the domain of g then domain of f g 0} = {x A B g(x) 0}. is A B {x g(x) = Example 2 Let f(x) = x 2 and g(x) = x 1. Find (fg)(x), ( f )(x), g g f their domains. and state 1
(e) Solution: (fg)(x) = x 2 x 1, and its domain is [1, ). ( f g )(x) = domain is (1, ). ( g f )(x) = x 1 x 2 and its domain is [1, ). x2 x 1 and its Composition of Functions: Given two functions f and g, composite function f g is defined as (f g)(x) = f(g(x)) Let A be the domain of f, B be the domain of g then domain of f g is {x B g(x) A}. Example 3 (a) Let f(x) = x + 1 x x+1, g(x) =. Compute f g, g f and their domains. x+2 (b) Let f(x) = 1 x 2, g(x) = cos (x). Compute the domains of f + g, f g, g f, f g and g f. Solution: (a) (f g)(x) = (x + 1)2 + (x + 2) 2 (x + 1)(x + 2) and its domain is (, 2) ( 2, 1) ( 1, ) (b) (g f)(x) = x2 + 1 + x x 2 + 1 + 2x and its domain is (, 1) ( 1, 0) (0, ) Domain of (f + g)(x) is (, 0) (0, ) Domain of f g is (, 0) (0, ) {(2k + 1)π k Z} 2 Domain of g f is (, 0) (0, ) Domain of f g is (, ) {(2k + 1) π k Z} 2 Domain of g f is (, 0) (0, ) Next we will see how to draw graphs of functions obtained from old functions. 2
Vertical and Horizontal Shifts: Suppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units upwards y = f(x) c, shift the graph of y = f(x) a distance c units downwards y = f(x c), shift the graph of y = f(x) a distance c units to the right y = f(x + c), shift the graph of y = f(x) a distance c units to the left Vertical and Horizontal Streching and Reflecting: Suppose c > 1. To obtain the graph of y = cf(x), stretch the graph of y = f(x) vertically by a factor of c y = 1 f(x), compress the graph of y = f(x) vertically by a factor of c c y = f(cx), compress the graph of y = f(x) horizontally by a factor of c y = f( 1 x), stretch the graph of y = f(x) horizontally by a factor of c c y = f(x), reflect the graph of y = f(x) about the x-axis y = f( x), reflect the graph of y = f(x) about the y-axis Example 4 The graph of y=f(x) is given below. Match each equation with its graph and give reasons for your choices. 1. y=f(x-4) 2. y=f(x)+3 3. y = 1 3 f(x) 4. y=-f(x+4) 5. y=2f(x+6) 3
Solution: 1. c 2. a 3. d 4. e 5. b Lets look at the graphs of f( x), f( x ) and f(2x) for a given f(x). 2 Next, let s see the relation between graphs of f(x), f(x) and f( x ). Note that { f(x) if f(x) 0 y = f(x) = f(x) if f(x) < 0 y = f( x ) = { f(x) if x 0 f( x) if x < 0 Example 5 Draw the graphs of y = sin (x) and y = sin ( x ) Solution: 4
Example 6 Graph the following functions: 1. y = 9 x 2. y = 1 x 1 3. y = 1 (x+1) 2 4. y = cos (2πx) Solution: 5
Chapter 2 Section 2.2: Limit of a Function and Limit Laws, Section 2.4: One-sided Limits We will see section 2.1 right before Chapter 3. Let f(x) = y be a function Idea of Limit: Can we find the value of f(x) near a certain x value, even when f is not defined at that certain value? In both graphs, g(x) and f(x) get closer to the value L as we approach x=a. In this graph as we approach x=0 the value of the function is different depending on from which side we are approaching. That is Definition 7 We write if we approach x = 0 from right f(x) = 1 if we approach x = 0 from left f(x) = 0 lim = L x a f(x) if f(x) gets arbitrarily close to L as we approach a from the left hand side. This is called the left hand side limit of f as x approaches a. We write lim = L x a + f(x) 6
if f(x) gets arbitrarily close to L as we approach a from the right hand side. This is called the right hand side limit of f as x approaches a. Definition 8 We write lim x a f(x) = L if lim x a f(x) = lim x a + f(x) = L and this is called the limit of f(x) as x approaches a. If lim x a f(x) lim x a + f(x), then we say limit of f(x) as x approaches a does not exist. Example 9 Graph of y=g(x) is given below. Find 1. lim x 0 g(x) 2. lim x 0 + g(x) 3. lim x 0 g(x) 4. g(0) 5. lim x 2 g(x) 6. lim x 2 + g(x) 7. lim x 2 g(x) 8. g(2) 9. lim x 4 g(x) 10. lim x 4 + g(x) 11. lim x 4 g(x) 12. g(4) Another case we say limit does not exist is when the function value become very large or very small near a point. 7
Definition 10 We write lim f(x) = x a if lim x a + f(x) = = lim x a f(x) if f(x) is arbitrarily large as x approaches a. We write lim f(x) = x a if lim x a + f(x) = = lim x a f(x) if f(x) is arbitrarily small as x approaches a. Note 11 Even though in the case lim x a f(x) = ± we say limit does not exist, it is usefull to distinguish between the previuos case where lim x f(x) lim x a f(x) and each limit is a number. We will come to this infinite limit case in Section 2.6. Example 12 Graph of y=f(x) is given below. Find 1. lim x 7 f(x) 2. lim x 3 f(x) 3. lim x 0 f(x) 4. lim x 6 f(x) 8
Limit Laws Suppose that c is a constant and limits lim x a f(x) and lim x a g(x) exists. Then 1. lim x a [f(x) + g(x)] = lim x a f(x) + lim x a g(x) 2. lim x a [f(x) g(x)] = lim x a f(x) lim x a g(x) 3. lim x a [cf(x)] = c lim x a f(x) 4. lim x a [f(x)g(x)] = lim x a f(x) lim x a g(x) 5. lim x a [ f(x) limx a f(x) ] = if lim g(x) lim x a g(x) x a g(x) 0 6. lim x a [f(x)] n = [lim x a f(x)] n 7. lim x a c = c 8. lim x a x = a 9. lim x a x n = a n where n is a positive integer. 10. lim x a n x = n a where n is a positive integer (if n is even we assume a > 0). 11. lim x a n f(x) = n lim x a f(x) where n is a positive integer (if n is even we assume lim x a f(x) > 0). Direct Substitution Property If f is a polynomial or a rational function and a is in the domain of f then lim f(x) = f(a) x a 9
Example 13 1. lim x 5 x+5 x+5 = 2. lim x 0 x 3 8 x 2 = 3. lim x 2 x 3 8 x 2 = 4. lim x 1 x+3 2 x 1 = x 5. lim x 0 sin (x) = x 6. lim x 0 x = Solution: 10