MAT135H1 Study Guide - Inverse Function, Inverse Relation

Consider the set X, and let F = {f|j: X rightarrow X), the set of functions from X to itself. A notable member of F is, of course, the identity function, id_x: X rightarrow X, defined by the formula: Forall x Element X. id_x(x) = x. In this exercise, we will study the concept of the inverse function. Recall that given two functions f, g Element F we have defined the operation of composition, which is denoted o and takes two functions f, g Element F and produces a new function as follows: f og(x) = f(g(x) for all x Element X. We will give the following definitions: (i) Def 1. A function g Element F if called an left inverse of a function f Element F go f = id_x. (ii) Def 2. A function h Element F if called an right inverse of a function f Element F if f o h = id_x. (iii) Def 3. A function l Element F if called an inverse of a function f Element F if l is simultaneously a left and a right inverse of f. Let LI(f) the predicate "f has a left inverse", RI(f) the predicate "f has a right inverse", and INV(f) the predicate "f has an inverse". Also let INJ(f) be the predicate "f is injective (one-to-one)", SUR(f) "f is surjective (onto)", and "BIJ(f) "f is bijective (one-to-one and onto). Express each of the following statements in predicate logic and write a proof. (i) A function f: X rightarrow X is injective if and only if it has a left inverse. (ii) A function f: X rightarrow X is surjective if and only if it has a right inverse. (iii) A function f: X rightarrow X is bijective if and only if it has an inverse. (iv) If the inverse g of a function f: X rightarrow X exists, it is unique. ("Unique" means the following: if f has another inverse g', then g = g'. Since the inverse of a function is unique, there is a special notation used for it: f ^-1).

Solve the equation. e x= e 10x+36 The function f(x) =4x/3x-5 is one-to-one. Find its inverse and check your answer. Suppose that H(x) =(1/2)x-16. What is H( - 6)? What is the point that corresponds to this value on the graph of H? If H(x) =16, what is x? What is the point that corresponds to this value on the graph of H? Find the zero of H. The point that corresponds to this value on the graph of H is (Type an ordered pair, but do not use commas in any individual coordinates. The point that corresponds to this value on the graph of H is (Type an ordered pair, but do not use commas in any individual coordinates. The zero of H is The function f(x) =6x/x+5 is one-to-one. Find its inverse and check your answer. Solve the equation. ex2=e15x.1/e54 What is the solution? The function f(x) =5/6+x is one-to-one. Find its inverse and check your answer. If f(x) = 4x2 + 1 and g(x) = 7x + a, find a so that the graph of fdegree g crosses the y-axis at 65. What is the value of a? If f(x) = 6x3-4x2 + x-8 and g(x)=3, find(f.g)(x) and (g.f)(x). What is (f.g)(x)? What is (g.f)(x)? Graph the function. Based on the graph, state the domain and the range and find any intercepts f(x)={-e-(x-2) if x=2 Choose the correct graph of f(x). The domain is (Type your answer in interval notation.) The range is (Type your answer in interval notation.) Find the intercepts, if there are any. Select the correct choice below and fill in any answer boxes within your choice. The intercept(s) is/are (Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer.) There are no intercepts.

Given f(x) = 3x2 - 2 and g(x) = 3 - 1/2 x2: find the following expressions. (f.g)(4) (g f)(2) (f.f)(l) (g.g)(0) (f . g)(4) = (Simplify your answer.) find the following expressions, (f.g)(4) (g.f)(2) (f.f)(l) (g.g)(0) (fog)(4) = (Type an exact answer, using radicals as needed. Simplify your answer.) Given f(x) = |x| and g(x) =9/x2+3, find the following expressions. (f.g)(4) (g f)(2) (f.f)(l) (g.g)(0) (f . g)(4) = (Type an integer or a simplified fraction.) For the function on the right determine whether the function is one-to-one. Is the function one-to-one? Yes No For the function below, determine whether the function is one-to-one. {(7,7),(8,8),(9,46),(10,51)} Is the function one-to-one? Yes No For the function below, determine whether the function is one-to-one. {(7,7),(8,8),(9,46),(10,51)} Is the function one-to-one? Yes No The graph of a function f is given. Use the horizontal-line test to determine whether f is one-to-one. Is f one-to-one? Choose the correct answer below. No Yes The graph of a function f is given. Use the horizontal-line test to determine whether f is one-to-one. Is f one-to-one? Choose the correct answer below. No Yes Find the inverse of the one-to-one function. State the domain and range of the inverse function. {(10,-8), (-13,-10), (-9,2), (-8,11), (-10,1)} Which of the following is the inverse function? {(-10,1), (-8,11), (-9,2), (-13,- 10), (10,-8)} {(-8,10), (-10,-13), (2,-9), (11,-8), (1,-10)} {(-8,-10), (-10,-8), (2,-9), (11,-13), (1,10)} {(-10,8), (13,10), (9,-2), (8,-11), (10,-1)} Consider the functions f(x) = - 3x + 7 and g(x) = -1/3(x - 7). Find f(g(x)). Find g(f(x)). Determine whether the functions f and g are inverses of each other. What is f(g(x))? f(g(x)) = (Simplify your answer.) Consider the functions f(x) = 3x - 15 and g(x) = x/3 + 5. Find f(g(x)). Find g(f(x)) Determine whether the functions f and g are inverses of each other. What is f(g(x))? f(g(x)) = (Simplify your answer.) Match the graph to one of the following functions. y = 6x y = 6-x y = -6x y = -6-x y = 6x_ 1 y = 6x-1 Which function is represented by the graph? (Type A, B, C, D, E, F, G, or H.) Begin with the graph of y = ex and use transformations to graph the following function. Determine the domain, range, and horizontal asymptote of die function. f(x) = 8e-x Choose the correct graph below. Solve the equation. 4x.2x2=162 x = (Use a comma to separate answers as needed.) If 4-x = 5;what does 42x equal? 42x = (Type a reduced fraction.)