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# College Algebra MTH 103

MSU

GPA 3.57

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This 9 page Class Notes was uploaded by Donny Graham on Saturday September 19, 2015. The Class Notes belongs to MTH 103 at Michigan State University taught by K. Park in Fall. Since its upload, it has received 18 views. For similar materials see /class/207305/mth-103-michigan-state-university in Mathematics (M) at Michigan State University.

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Date Created: 09/19/15

Section 27 Inverse Functions De nition of the Inverse of a Function Let f and g be two functions such that fg X X for every x in the domain of g and g fX X for every x in the domain off The function g is the inverse of the function f and is denoted by I read inverse Thus f f1X X and f1 fX X The domain of f is equal to the range of f1 and range of f is equal to the domain of 1 1 Example 1 Show that each function is the inverse of the other fo4x 5 and gap Finding the Inverse of a Function Example 2 Find the inverse of 3X 2 Step 1 Replace with y Step 2 Interchange x and y Step 3 Solve for y Step 4 Replacey with f1X X4 2X 339 Example 3 Find the inverse of Example 4 Find the inverse of 11X A 4 The Horizontal Line Test and One to One Functions A function f has an inverse I if there is no horizontal line that intersects the graph of the function f at more than one point This means that if Xi 7E X2 then 1quot213 7E fX2 Such a function is called a one to one function Thus only onetoone functions have inverse functions Example 5 Which of the following graphs represent functions that have inverse functions ie onetoone functions y y y 7x 7x a b C Section 35 Rational Functions and Their Graphs Rational Functions Rational functions are quotient of polynomial functions Rational functions can be expressed as fx qx where p and q are polynomial functions and qx 2 0 The domain of a rational function is the set of all real numbers except the xValues that make the denominator zero Example 1 Find the domain of each rational function 2 x 25 x x 5 21 fx b 00 hx x2 25 c x225 Reciprocal function 1 The Reciprocal function is defined by fx x Arrow Notation Symbol Meaning x gt 0 x approaches a from the right x gt a x approaches a from the left x gt 00 x approaches in nity that is x increases without bound x gt 00 x approaches negative in nity that is x decreases without bound 1 Veltical Asymptotes of Rational Functions The line x a is a vertical asymptote 0f the graph of a function f if increases or decreases without bound as x approaches a As x gt a fx gtoo or 00 Finding Vertical Asymptotes 17x f I M W and a is a zero of qx the denominator then x a is a vertical asymptote 0f the graph of Example 2 Find the vertical asymptotes algebraically if any of the graph of each rational function a fx is a rational function in which px and qx have no common factors x x2 1 x l x l b x hx g x2 1 c x21 Horizontal Asymptotes of Rational Functions The line y b is a horizontal asymptote 0f the graph of a function f if approaches b as x increases of decreases without bound As x gtoo or 00 fx gtb Finding Horizontal Asymptotes Let f be the rational function given by x f lame bm71xm71b1xb0 The degree of the numerator is n The degree of the denominator is m 1 If n lt m the xaXis or y 0 is the horizontal asymptote 0f the graph of VL 7171 ax 0 X axa a 0bm 0 a 2 If n m the line y b is the horizontal asymptote of the graph of 3 If n gt m the graph of f has no horizontal asymptote Example 3 Find the horizontal asymptote algebraically if any of the graph or each rational function 9x2 9x 9x3 x b x hx a f 3x21 gm 3x21 c 3x21 Using Transformations to Graph Rational Functions Some rational functions can be graphed using transformations HSRV of two common 1 l graphs fx and gx 2 x x l 1 Example 4 Use the graph of 2 to graph gx 2 1 x x 2 Graphing Rational Functions 3x x Step 1 Determine symmetry f x even function yaXis symmetry Example 5 Graph fx f x fx odd function origin symmetry Step 2 Find the yintercept by evaluating Step 3 Find the xintercepts by solving the equation px 0 Step 4 Find any vertical asymptote by solving the equation qx 0 Step 5 Find the horizontal asymptote using the rule Step 6 Plot points between and beyond each x intercept and vertical asymptote Step 7 Graph the function Chapter 26 Combinations of Functions Composite Functions Finding a Function s Domain The domain of a function f is the largest set of real numbers for which the value of f x is a real number Exclude the numbers which cause Division by Zero and a Square Root of a Negative Number Example 1 Find the domain of each function a fX A 7X 2 4X3 b 7 M A 9 c 11X 4X 16 E x2 4 d Px The Algebra of Function lSum f l39gXX fXgX 2 Difference f gX fX gX 3 Product x fXgX 4 Quotient IXY Q provideng at 0 g gX Each domain is the intersection of domains of f and g ExampleZ Let fX X 4 and gX A 3X 2 Find the following functions a fgX b fgXX 0 X L x and Example 3 Let fX X 2 and gX X 4 Find f gX and the domain of f g Composite Functions The composition of the function f with g is W300 f5729 This can be read f of g of x or f composed with g of x The domain of the composite function W is the set of all x such that 1 x is the domain of g and 2 gX is in the domain off Example 4 Given fX 5X 3 and gX A X 2 nd each 0fthe following composite functions a N39 QXX b NED XX

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