Math questions

In section 2.7, we will modify the equations of the “basic” functions and see what effect these modifications have on the graphs of the modified functions. Next, we will also learn how to classify functions as odd, even or neither And finally, we will test a given function for possible symmetries of its graph about the x-axis, the y axis, or about the origin Note: For this Section, I recommend that you FIRST study the materials from the text and then supplement your understanding of the topic by going over the examples given below in these notes and also studying th The possible modification that we can do to the graph of y = f(x) are: • Shift (or translate) the graph horizontally left or right • Shift(or translate) the graph vertically up or down • Reflect the graph about the X axis • Reflect the graph about the Y axis • Vertically stretch or shrink the graph • Horizontally stretch or shrink the graph The textbook has covered these with examples in great details and has also summarized all the modifications in a table form. The title of this table is ” Summary of Graphing Techniques” and it is given at the very end of this section, just before the exercise set. Use the text to first LEARN these modifications. You need to know them well enough so that given any function equation you can do the following: • • Identify the corresponding basic function and draw its graph Identify all modifications done to a basic function and sketch the corresponding graph. Note: These modifications do NOT change the shape of the original graph Objective Given the equation of a function, to be able to identify the corresponding ‘basic’ function and then graph the equation. Examples Below are listed some sample problems. As all the graphs in this set have the same basic shape, their graphs have not been included. Summary Table refers to the Table titled ” Summary of Graphing Techniques” given at the end of this section in the text Sample Problems The ‘basic function’ for each of these problems is y = f(x) = |x|. The graph of y = |x| is shaped like the letter ‘V’ with its vertex at (0,0) #1. y = |x – 3| Here we have changed the equation y = |x| by replacing “x” with “(x – 3)” Therefore we shift the graph of y = |x| by three units to the right . [Summary table – Property 4] The shifted vertex is at (3,0) #2. y = – |x| In this case, since “f(x) ” is replaced with “[-f(x)]”. Therefore the graph of f(x) gets reflected about the X axis and we end up with an inverted “v” shape. [Summary table-Property 7]. The vertex remains at (0,0) #3. y = |x| – 3 Here, we have replaced “f(x)” with “[f(x) – 3]” ⟹ Summary table-Property 2 applies and the original graph shifts down three units. The shifted vertex is now at the point (0,-3) #4. y = |x + 4| + 1 In this example, we need Summary table-Properties 1 and 3. From Property 3, since “x” is replaced by “[x + 4]”, the graph shifts to the left four units. In addition, because “f(x)” is changed to “[f(x) + 1]”, the graph also moves up one unit. The new vertex is now at the point (-4, 1) Note: It is always a good idea to also find the x and the y intercepts (if possible) of the modified equation before graphing it Even and Odd Functions Definition -1 A function f is called an Even Function if f(-x) = f(x) for all x in the domain of f Definition-2 A function is said to be an Odd Function if f(-x) = -f(x) for all x in the domain of f. Symmetry Tests Thirteenth Ed: Refer to Page 284 of the text for the table titled “Summary of Tests for Symmetry” Twelfth Ed: Refer to Page 265 of the text for the table titled “Summary of Tests for Symmetry” Questions Q) The function y = x has been modified to the function Without graphing a Is there Is there ⑨ Is a a horizontal vertical there a , each of answer translation ? translation ? reflection ? If Find the coordinates of f Find the coordinates of the If the f(X) : -(x + 2)” + 1 following . Yes, how in much If Yes, how much , about which yes the : axis in ! y-intercept of f(x) X-intercept of FCA) what what ? direction ? direction We will begin this section with a brief review of function notations using examples. Next we will learn how to add, subtract, multiply or divide two given functions. In addition, we will also define the Difference Quotient of a function. And finally, we will learn how to perform composition of two given functions. Function Notation Problem: Given a function y = f(x), find f(t), where ‘t’ is some algebraic or numeric expression Method: Replace ‘x’ by ‘t’ on the right side in the definition of f Examples of Function Notation #1. Problem: Given f(x) = 3x – 7. Find f(4), f(-6) and f(a). Assume ‘a’ is any real number Solution: f(4) = 3(4) – 7 = 12 – 7 = 5 f(-6) = 3(-6) – 7 = -18 – 7 = -25 f(a) = 3(a) – 7 = 3a – 7 #2. Problem: Given f(x) = x2 + 3, find (a) f(-2) (b) f(x + 2). Solution: (a) f(-2) = (-2)2 + 3 = 4 + 3 = 7 (b) f(x + 2) = (x + 2) 2 + 3 = (x2 + 4x + 4) + 3 or f(x + 2) = x2 + 4x + 7 For definitions of addition, subtraction, multiplication and division of functions, refer to the text for the table titled ” Operations on Functions and Domains” on Page 297 (Thirteenth Ed.) Solved examples based upon these operations are in the attached solution set file. Difference Quotient Let f(x) be a given function. Then the difference quotient of f(x) is given by the expression: f(x+h)−f(x)h Steps Needed to Find the Difference Quotient of a Given Function f(x) Step-1: Find f(x + h) and simplify the answer Step-2: Place this value in the numerator of the difference quotient in place of f(x+ h) Step-3: Simplify Hint: Make sure you distribute the negative sign over f(x) if f(x) has two or more terms. If your work is correct, after simplification you will find “h” to be a common factor between the numerator and denominator. This common factor must be reduced to simplify the answer. An Example of Difference Quotient: Problem: Given f(x) = 2x + 3, find (a) f(x + h) (b )f(x+h)−f(x)h Solutions: (a) f(x) = 2x + 3, ⟹ (b) f(x+h)−f(x)h f(x + h) = 2(x + h) + 3⟹f(x + h) = 2x + 2h + 3 = (2x+2h+3)−(2x−3)h = 2x+2h+3−2x−3 h = 2h h ⟹ f(x+h)−f(x) h =2 Additional solved examples on difference quotient are included in the solution set attached at the end of this page Composition of Functions and Domains If f and g are functions, then the composition of f and g is defined as (f ∘ g)(x) = f[g(x)] The domain of the composite function f ∘ g is given by the set : {x| x is in the domain of g and g(x) is in the domain of f} Example: Let f(x) = 3x, and g(x) = x2 Find (f ∘ g)(x) Solution: (f ∘ g)(x) = f[g(x) ] Following the order of operations, we first simplify the inner parenthesis = f[x2 ] From the definition of g(x) = 3(x2) From the definition of f(x) Therefore, (f ∘ g)(x) = 3×2 Property-1 (f∘g)(x) need not be the same as (g∘f)(x) Example: Let us once again take f(x) = 3x and let g(x) = x2 . Solution: Proceeding as before, we get (g ∘ f)(x) = g[f(x)] = g[3x] = (3x) 2 ⟹ (g ∘ f)(x) = 9×2 But as shown earlier, (f ∘ g)(x) = 3×2 Since 3×2 ≠ 9×2, ⟹ (g ∘ f)(x) ≠(f∘g)(x) Questions ④) Given f(x) Find a a) a each of = x + the 1 and g(x) = x2- 1 following and simplify the answers. (fog)(x) iven g Show all . steps b(go f) (x) f(x) = 2×2 -1 fC 1 Find b – each of the simplify the answer following and – f(x + 2) c f(x + n) – f(x) h ④) 2 given f(x) a(f – = g)(x) 5* + 4 and g(x) = * Find each b(z)(x) of the following · n FO functions & state interval notation ne n its the . Questions Q a 1 2 Sketch the . graph of y Apply translation . b 1 C Use . Use the to = x the graph graph -(x) to determine of if part the abgebraic method to find the equation of i to function the draw is inverse a the 11 graph of the function function . Give reasons function y = for . Show all steps. f(x) your f(x) =+ 1 , answer Clearly label the coordinates of the Do not graph the Cy intercepts of – – (X) on the graph . Hint : X function .

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