2 Objectives:Find the domain of a function.Combine functions using the algebra of functions, specifying domains.Form composite functions.Determine domains for composite functions.Write functions as compositions.
3 Finding a Function’s Domain If a function f does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of f(x) is a real number. Exclude from a function’s domain real numbers that cause division by zero and real numbers that result in a square root of a negative number.
4 Example: Finding the Domain of a Function Find the domain of the functionBecause division by 0 is undefined, we must exclude from the domain the values of x that cause the denominator to equal zero.We exclude 7 and – 7 from the domain of g.The domain of g is
5 The Algebra of Functions: Sum, Difference, Product, and Quotient of Functions Let f and g be two functions. The sum f + g, the difference, f – g, the product fg, and the quotientare functions whose domains are the set of all real numbers common to the domains of f anddefined as follows:1. Sum:2. Difference:3. Product:4. Quotient:
6 Example: Combining Functions Let and Find each of the following:a.b. The domain ofThe domain of f(x) has no restrictions.The domain of g(x) has no restrictions.The domain of is
7 The Composition of Functions The composition of the function f with g is denotedand is defined by the equationThe domain of the composite function is the set of all x such that1. x is in the domain of g and2. g(x) is in the domain of f.
8 Example: Forming Composite Functions Given and find
9 Excluding Values from the Domain of The following values must be excluded from theinput x:If x is not in the domain of g, it must not be in the domain ofAny x for which g(x) is not in the domain of f must not be in the domain of
10 Example: Forming a Composite Function and Finding Its Domain Given andFind
11 Example: Forming a Composite Function and Finding Its Domain Given andFind the domain ofFor g(x),ForThe domain of is
12 Example: Writing a Function as a Composition Express h(x) as a composition of two functions:If and then
13 Objectives:Verify inverse functions.Find the inverse of a function.Use the horizontal line test to determine if a function has an inverse function.Use the graph of a one-to-one function to graph its inverse function.Find the inverse of a function and graph both functions on the same axes.
14 Definition of the Inverse of a Function Let f and g be two functions such thatf(g(x)) = x for every x in the domain of gandg(f(x)) = x for every x in the domain of fThe function g is the inverse of the function f and is denoted f –1 (read “f-inverse). Thus, f(f –1 (x)) = x andf –1(f(x))=x. The domain of f is equal to the range off –1, and vice versa.
15 Example: Verifying Inverse Functions Show that each function is the inverse of the other:andverifies that f and g are inverse functions.
16 Finding the Inverse of a Function The equation for the inverse of a function f can be found as follows:1. Replace f(x) with y in the equation for f(x).2. Interchange x and y.3. Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function.
17 Finding the Inverse of a Function (continued) The equation for the inverse of a function f can be found as follows:4. If f has an inverse function, replace y in step 3 byf –1(x). We can verify our result by showing thatf(f –1 (x)) = x and f –1 (f(x)) = x
18 Example: Finding the Inverse of a Function Find the inverse ofStep 1 Replace f(x) with y:Step 2 Interchange x and y:Step 3 Solve for y:Step 4 Replace y with f –1 (x):
19 The Horizontal Line Test for Inverse Functions A function f has an inverse that is a function, f –1, if there is no horizontal line that intersects the graph of the function f at more than one point.
20 Example: Applying the Horizontal Line Test Which of the following graphs represent functions that have inverse functions?a. b.Graph b represents a function that has an inverse.
21 Graphs of f and f – 1The graph of f –1 is a reflection of the graph of f about the line y = x.
22 Example: Graphing the Inverse Function Use the graph of f to draw the graph of f –1
23 Example: Graphing the Inverse Function (continued) We verify our solution by observing the reflection of the graph about the line y = x.