Fry Texas A&M University!! Math 150!! Chapter 4E!! Fall 2015! 121 Chapter 4E - Combinations of Functions 1. Let f (x) = 3 x and g(x) = 3+ x a) What is the domain of f (x)? b) What is the domain of g(x)? c) ( f + g)(x) = d) What is the domain of ( f + g)(x)? e) ( f g)(x) = f) What is the domain of ( f g)(x)? g) ( fg)(x) = h) What is the domain of ( fg)(x)? i) f g (x) = j) What is the domain of f g (x)?
Fry Texas A&M University!! Math 150!! Chapter 4E!! Fall 2015! 122 Practicing with function notation: 2. Let f (x) = x 2 + x Determine a) f (2) b) f ( 3) c) f ( ) d) f (x 2 ) e) f (4x) f) f (x + h) Function Composition Definition For functions f (x) and g(x), the composition of functions ( f! g)(x) is defined as 3. Let f (x) = x 2 9 and g(x) = 1 x + 5 a) ( f! g)( 3) = b) (g! f )( 3) = c) ( f! g)(x) = d) (g! f )(x)=
Fry Texas A&M University!! Math 150!! Chapter 4E!! Fall 2015! 123 4. Here are the graphs of two functions. If the line is the graph of f (x) and the parabola is the graph of g(x), sketch ( f + g)(x) 5. Using the same graph and functions, determine a) ( f! g) ( 4) b) ( g! f )( 4) c) ( f! g) ( 0) d) ( g! f )( 0)
Fry Texas A&M University!! Math 150!! Chapter 4E!! Fall 2015! 124 Extra Problems: Text: 3, 8-11 14. (4 points) Let f (x) = 5x +1 and g(x) = x 2 + 3 then a) ( f! g) ( 1) =!! b) ( f + g) ( 3) = 15. (4 points) Let f (x) = 2x + 5 and g(x) = 6x + 35 then a) ( f! g) ( x) =!! b) f g x ( ) = Let g(x) = 1 1 x 2 and h(x) = 1 x. ( ) x Simplify g! h ( ). a) x 2 ( 1 x) 1+ x ( ) b) x 2 ( x 1) x +1 ( ) c) 1 x 1 x ( ) 1+ x ( ) d) 1 x 2 1 e) None of these
Fry Texas A&M University!! Math 150!! Chapter 4F!! Fall 2015! 125 Chapter 4F Inverse Functions Let f (x) = x 3 3 1 and g(x) = x +1! Determine the following: a) ( f! g)(0)!!!!!!! b) (g! f )(0) c) ( f! g)(x)!!!!!! d) (g! f )(x) In basic terms two functions are said to be if. The notation for a function s inverse is It is important to understand that f 1 (x) does NOT represent In our example above, g(x) = f 1 (x) but we could also say One to one functions: Before we write a formal definition for an inverse function, we need to understand what it means for a function to be one-to-one. If f (x) is a function and if x 1 = x 2 can you be sure that f (x 1 ) = f (x 2 )? Why? For a function to have an inverse it has to be. f (x) is a one-to-one function if no two elements in the domain correspond to the same element in the
Fry Texas A&M University!! Math 150!! Chapter 4F!! Fall 2015! 126 The function f (x) = x 2 is not 1-1 because We see that there are two distinct elements in the domain that correspond to the By definition, f (x) is a one-to-one function if f (x 1 ) = f (x 2 ) implies that The function g(x) = x 3 is. If x 1 3 = x 2 3 then A function that is 1-1 will pass the That means that any line will intersect the graph of a one to one function at most. Notice how the graph of f (x) = x 2 the horizontal line test while the graph of g(x) = x 3. Definition: Let f (x) be a function with domain A and range B. Then the inverse function of f (x) is denoted has domain and range It is defined by f 1 (y) = x iff This means that if (a, b) is on the graph of f (x), then will be on the graph of f 1 (x). It is also important to note that f 1 ( f (x)) = for every x in and f ( f 1 (x)) = for every x in
Fry Texas A&M University!! Math 150!! Chapter 4F!! Fall 2015! 127 1. Consider f (x) = 1 x +1 a) What is the domain of f (x)? b) What is the range of f (x)? c) Show that f (x) a 1-1 function. To do this, we must show that Start by assume 1 x 1 +1 = 1 x 2 +1 Remember if two numbers are equal, their reciprocals are equal, so d) Find f 1 (x)! Step 1: Write the function with y instead of f (x) Step 2: Swap x and y Step 3: Solve for y.
Fry Texas A&M University!! Math 150!! Chapter 4F!! Fall 2015! 128 So f 1 (x) = e) What is the domain of f 1 (x)? (Notice that the domain of f 1 (x) is the same as ) f) What is the range of f 1 (x)? (Notice that the range of f 1 (x) is the same as ) g) Verify that f 1 ( f (x)) = f ( f 1 (x)) = x h) Sketch both f (x) and f 1 (x) on the same graph at the top of page 3
Fry Texas A&M University!! Math 150!! Chapter 4F!! Fall 2015! 129 2. Consider f (x) = 4 2x!!! a) What is the domain of f (x)? b) What is the range of f (x)?!!!! c) Show that f (x) a 1-1 function: To do this, we must show that Start by assuming that 4 2x 1 = 4 2x 2 Remember if two numbers are equal, their squares are equal, so d) Find f 1 (x) Step 1: Write the function with y instead of f (x) Step 2: Swap x and y Step 3: Solve for y. e) What is the domain of f 1 (x)? (Notice that the domain of f 1 (x) is the same as ) f) What is the range of f 1 (x)? (Notice that the domain of f 1 (x) is the same as ) h) Sketch both f (x) and f 1 (x) on the same graph
Fry Texas A&M University!! Math 150!! Chapter 4F!! Fall 2015! 130 When a function is not 1-1, we sometimes restrict the of f (x) so that the function is then 1-1. 3) Consider f (x) = x 2. If we restrict the domain of f (x) to (, 0], then f (x) is 1-1 on this restricted domain a) On the restricted domain, what is the range of f (x)? b) Find f 1 (x) Step 1: Write the function with y Step 2: Swap x and y Step 3: Solve for y. c) What is the domain of f 1 (x)? d) What is the range of f 1 (x)? f e) Sketch both f (x) and f 1 (x) on the same graph.
Fry Texas A&M University!! Math 150!! Chapter 4F!! Fall 2015! 131 It is important to note that the graph of f 1 (x) can be found by 4) Given that f (x) = x +1 x 2 function. is a one-to-one function on (, 2) (2, ), find its inverse Domain of f (x)! Range of f (x) Domain of f 1 (x)! Range of f 1 (x) Extra Problems:! Text: 3, 8-11
Fry Texas A&M University!! Math 150!! Chapter 4F!! Fall 2015! 132 1. Circle the best answer. A function is one-to-one if a) f ( x) = f (x) b) f ( x) = f (x) c) f (x 1 ) = f (x 2 ) implies that x 1 = x 2 d) x 1 = x 2 implies that f (x 1 ) = f (x 2 ) e) it passes the vertical line test! 2. If f (x) = 4 x 3x + 2.! a) Then f 1 (x) = a) 3x + 2 4 x b) 3x + 2 x 4 c) 4 2x 3x +1 d) 2x 4 3x +1 e) None of these In interval notation state b) the domain of f (x) c) the range of f (x) d) the domain of f 1 (x)! e) the range of f 1 (x)
Fry Texas A&M University!! Math 150!! Chapter 5A!! Fall 2015! 133 Chapter 5A Polynomials Polynomial Degree Leading Term Constant Term f (x) = 4x 3 + 2x 5 g(x) = 17x 5 + 6x 3 h(x) = πx 6 17 c(x) = 14 p(x) = p n x n + p n 1 x n 1 +!+ p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers, the p i 's are real numbers and p n 0 r(x) = 2 + 3x 5x 4 We ve already graphed polynomials of degree 1 (lines) and polynomials of degree 2 (parabolas). We have also graphed some polynomials of degree 3 like these: Cubics: f (x) = x 3!!!!!!!! g(x) = (x 2) 3 Leading Term: Constant Term: Roots: Leading Term: Constant Term: Roots: As x, x 3 As x, (x 2) 3 As x, x 3 As x, (x 2) 3
Fry Texas A&M University!! Math 150!! Chapter 5A!! Fall 2015! 134 h(x) = 2(x + 3)(x 1) 2 Leading Term: Constant Term: Roots: As x, 2(x + 3)(x 1) 2 As x, 2(x + 3)(x 1) 2 Now we would like to solve 2(x + 3)(x 1) 2 > 0 (because when 2(x + 3)(x 1) 2 > 0, the graph of h(x) will be the x-axis.) Draw a number line and plot the zeros list the factors to create a sign table determine the sign of each factor use the signs of the factors to determine the sign of the product h(x) > 0 when so the graph of h(x) is above the x-axis when
Fry Texas A&M University!! Math 150!! Chapter 5A!! Fall 2015! 135 p(x) = 1 4 (x + 2)(x 2)(3x 4) Leading Term: Constant Term: Roots: As x, 1 4 (x + 2)(x 2)(3x 4) As x, 1 4 (x + 2)(x 2)(3x 4) Solve p(x) > 0. Draw a number line and plot the zeros list the factors to create a sign table determine the sign of each factor use the signs of the factors to determine the sign of the product p(x) > 0 when so the graph of p(x) is above the x-axis when
Fry Texas A&M University!! Math 150!! Chapter 5A!! Fall 2015! 136 True or False: Some cubics never intersect the x-axis. Some cubics intersect the x-axis in exactly one place. Some cubics intersect the x-axis in exactly two places. Some cubics intersect the x-axis in three places. Some cubics intersect the x-axis in four places. Quartics f (x) = x 4 +1!!!!!!! g(x) = 1 4 (x 2) 4 Leading Term: Leading Term: Constant Term: Roots: As x, x 4 +1 As x, x 4 +1 Constant Term: Roots: As x, 1 (x 2) 4 4 As x, 1 (x 2) 4 4
Fry Texas A&M University!! Math 150!! Chapter 5A!! Fall 2015! 137 h(x) = x 3 (2x + 7) Leading Term: Constant Term: Roots: As x, x 3 (2x + 7) As x, x 3 (2x + 7) Solve h(x) = x 3 (2x + 7) > 0 p(x) = (x 2 1)(x + 2) 2 Leading Term: Constant Term: Roots: As x, p(x) = (x 2 1)(x + 2) 2 As x, p(x) = (x 2 1)(x + 2) 2 Solve p(x) = (x 2 1)(x + 2) 2 > 0
Fry Texas A&M University!! Math 150!! Chapter 5A!! Fall 2015! 138 Polynomials of Higher Degree In general, to understand the behavior of a polynomial, 1. Plot the y-intercept 2. Find the zeros, 3. Determine where the polynomial is positive and negative because this will tell you where the graph is above and below the x-axis. 4. Determine the behavior of the polynomial for large positive values of x and for large negative values of x. The behavior of the polynomial at these extremes will be dominated by the leading term, the term with the highest power of x. f (x) = x 5!!!!!!! g(x) = x 5!!! Leading Term: Constant Term: Roots: Leading Term: Constant Term: Roots: x 5 > 0 x 5 > 0 As x, x 5 As x, x 5 As x, x 5 As x, x 5
Fry Texas A&M University!! Math 150!! Chapter 5A!! Fall 2015! 139 h(x) = (3x 1) 2 (x 2)(x +1)(x + 3)!!!!!!! Leading Term: Constant Term: Roots: Solve h(x) = (3x 1) 2 (x 2)(x +1)(x + 3) > 0 As x, h(x) = (3x 1) 2 (x 2)(x +1)(x + 3) As x, h(x) = (3x 1) 2 (x 2)(x +1)(x + 3)
Fry Texas A&M University!! Math 150!! Chapter 5A!! Fall 2015! 140 f (x) = x(x 3) 2 (2x + 5) 2 Degree of the polynomial: Leading Term: Constant Term: Zeros: Solve: x(x 3) 2 (2x + 5) 2 > 0 As x, x(x 3) 2 (2x + 5) 2 As x, x(x 3) 2 (2x + 5) 2
Fry Texas A&M University!! Math 150!! Chapter 5A!! Fall 2015! 141 The end behavior of a polynomial can be determined by examining the end behavior of the True or False: All linear functions cross the x-axis. True or False: All quadratic functions cross the x-axis. True or False: All cubic functions cross the x-axis. True or False: All quartic functions cross the x-axis. True or False: All quintic functions cross the x-axis. True or False: All polynomials functions of even degree cross the x-axis. True or False: All polynomials functions of an odd degree cross the x-axis. Extra Problems: Text:! 1-8! 1. f (x) = x 2 (1 x 3 )!! As x, f (x) a) b) c) 1 d) 0 e) 1 2.! g(x) = x 2 (1 x 2 )! As x, g(x)! a) b) c) 1 d) 0 e) 1 3 a) If g(x) = x 4 + 5, then as x, g(x) b) If h(x) = x 5 + 7, then as x, h(x)
Fry Texas A&M University!! Math 150!! Chapter 5A!! Fall 2015! 142 4. Let f (x) = (x 2 + 2)(x 1) 2 Determine the following.! If there are none, write NONE. Intercepts are points so state their x and y coordinates. a) y - intercept(s)!! b) x - intercept(s) c) Solve (x 2 + 2)(x 1) 2 0!! d) Sketch the graph of f (x) = (x 2 + 2)(x 1) 2 (This sketch does not have to be perfect, but it should include the information you found above as well as demonstrating the end behavior of f (x) = (x 2 +1)(x 2) 2 )