Mathematics for Business and Economics - I Chapter 5. Functions (Lecture 9)
Functions The idea of a function is this: a correspondence between two sets D and R such that to each element of the first set, D, there corresponds one and only one element of the second set, R. The first set is called the domain, and the set of corresponding elements in the second set is called the range. Notation: if y is a function of x, we write y = f(x) Other common symbols for functions include but are not limited to g, h, F, G
Consider our function f x 2 ( ) x 2 What does f (-3) mean? Replace x with the value 3 and evaluate the expression 2 f ( 3) ( 3) 2 The result is 7. This means that the point (-3,7) is on the graph of the function.
EXAMPLE 1 Evaluating a Function Let g be the function defined by the equation y = x 2 6x + 8. Evaluate each function value. a. g3 Solution b. g 2 d. ga 2 e. gx h a. g3 3 2 63 8 1 b. g2 c. g 1 2 2 2 62 8 24
EXAMPLE 1 Evaluating a Function Solution continued c. g 1 2 1 2 2 6 1 2 8 21 4 d. ga 2 a 2 2 6a 2 8 e. gx h a 2 4a 4 6a 12 8 a 2 2a x h 2 6x h 8 x 2 2xh h 2 6x 6h 8
If the domain of a function that is defined by an equation is not explicitly specified, then we take the domain of the function to be the largest set of real numbers that result in real numbers as outputs.
Consider f ( x) 3x 2 f (0)? f ( 0) 3( 0) 2 2 which is not a real number. Question: for what values of x is the function defined? Answer: f ( x) 3x 2 is defined only when the radicand (3x-2) is equal to or greater than zero. This implies that x 2 3 Therefore, the domain of our function is the set of real numbers that are greater than or equal to 2/3.
Example1: Find the domain of the function 1 f ( x) x 4 2 Answer: xx 8, [8, ) Example : Find the domain of 1 f( x) 3x 5 In this case, the function is defined for all values of x except where the denominator of the fraction is zero. This means all real numbers x except 5/3.
EXAMPLE 3 Finding the Domain of a Function Find the domain of each function. a. f x c. hx 1 1 x 2 1 x 1 b. gx Solution a. f is not defined when the denominator is 0. 1 x 2 0 x 1 x d. Pt 2t 1 Domain: {x x 1 and x 1}, 1 1,1 1,
EXAMPLE 3 Finding the Domain of a Function Solution continued b. gx x The square root of a negative number is not a real number and is excluded from the domain. Domain: {x x 0}, [0, ) 1 c. hx x 1 The square root of a negative number is not a real number and is excluded from the domain, so x 1 0. However, the denominator 0.
EXAMPLE 3 Finding the Domain of a Function Solution continued So x 1 > 0 so x > 1. Domain: {x x > 1}, or (1, ) d. Pt 2t 1 Any real number substituted for t yields a unique real number. Domain: {t t is a real number}, or (, )
Example #2a (p.80) Find the domain of f(x) = x/(x 2 x 2) The domain would be the set of all real numbers except those values of x which set the denominator equal to zero These values are found by factoring (x 2 x 2) = (x + 1)(x - 2) x = -1, 2 So the domain is the set of all real numbers, except x =-1, 2
Two functions, f and g are equal (f = g) if The domain of f is equal to the domain of g For every x in the domain of f and g, the values of the two functions are the same; that is f(x) = g(x)
Which of the following functions are equal f(x) = (x + 2)(x + 1)/(x 1) g(x) = x + 2 h(x) = x + 2 Domains of g, h, the set of all real numbers and are equal, but the domain of f is the set of all real numbers except x = 1
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2 A polynomial function of degree n is a function of the form f x a n x n a n1 x n1... a 2 x 2 a 1 x a 0, where n is a nonnegative integer and the coefficients a n, a n 1,, a 2, a 1, a 0 are real numbers with a 0.
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2.3 Combinations of Functions (i) Sum (ii) Difference f g f g x f x gx x f x gx (iii) Product fgx f x g x (iv) Quotient (v) f g, x f x g x g x 0.
EXAMPLE 1 Combining Functions
EXAMPLE 2 Combining Functions f(x) = x 2, g(x) = 3x, find; i. f(x) + g(x) = x 2 + 3x ii. f(x).g(x) = 3x 3 iii. f(x) g(x) = x 2 3x iv. f(x)/g(x) = x 2 /3x = x/3 v. cf(x) = cx 2
EXAMPLE 3 Combining Functions Solution continued f x x 2 6x 8 and g x x 2 b. f gx f x gx x 2 6x 8 x 2 x 2 7x 10 c. fgx x 2 6x 8 x 2 x 3 2x 2 6x 2 12x 8x 16 x 3 8x 2 20x 16
EXAMPLE 3 Combining Functions Solution continued f x x 2 6x 8 and g x x 2 d. f g, x f x g x gx 0 x2 6x 8, x 2 0 x 2 x 2x 4, x 2 x 2
If f and g are two functions, the composition of function f with function g is written as f og and is defined by the equation f ogx f g x, where the domain of f og consists of those values x in the domain of g for which g(x) is in the domain of f.
EXAMPLE 1 Evaluating a Composite Function Let f x x 3 and gx x 1. Find each of the following. a. f og1 b. g of 1 c. f of 1 d. g og1 Solution a. f og1 f g 1 f 2 2 3 8
EXAMPLE 1 Evaluating a Composite Function Solution continued f x x 3 and gx x 1 b. g of 1 gf 1 g1 1 1 2 c. f of 1 f f 1 f 1 1 3 1 d. g og1 gg 1 g0 0 1 1
EXAMPLE 2 Finding Composite Functions Let f x 2x 1 and gx x 2 3. Find each composite function. a. f ogx Solution a. f og b. g of x f g x x c. f of x f x 2 3 2x 2 3 1 2x 2 6 1 2x 2 5
EXAMPLE 2 Finding Composite Functions Solution continued f x 2x 1 and gx x 2 3. b. g of x gf x g2x 1 2x 1 2 3 4x 2 4x 2 c. f of x f f x f 2x 1 22x 1 1 4x 3
EXAMPLE 3 Finding the Domain of a Composite Function Let f x x 1 and gx 1 x. a. Find f og1. b. Find g of 1. c. Find f og x d. Find g of x Solution a. f og and its domain. and its domain. 1 f g 1 f 1 1 1 0
EXAMPLE 3 Finding the Domain of a Composite Function Solution continued f x x 1 and gx 1 x b. g of 1 gf 1 c. f ogx f gx g0 not defined f Domain is (, 0) U (0, ). 1 x 1 x 1 d. g of x gf x gx 1 1 x 1 Domain is (, 1) U ( 1, ).
EXAMPLE 4 Decomposing a Function 1 Let H x Show that each of the 2x 2 1. following provides a decomposition of H(x). a. Express H x, as f gx where f x 1 x and g x 2x 2 1. b. Express H x, as f gx where f x 1 x and gx 2x2 1.
EXAMPLE 4 Decomposing a Function Solution f 2x 2 1 a. f g x 1 2x 2 1 H x b. f g x f 2x 2 1 1 2x 2 1 H x