2. Function Theory 2.1 Introduction to Functions 2.2 Algebra of - PowerPoint PPT Presentation

2. Function Theory

2.1 Introduction to Functions 2.2 Algebra of Functions 2.3 Domain and Range of a Function 2.4 Inverse Functions

2.1 Introduction to Functions

What is a Function? • Functions are mathematical objects that send an input to a unique output. • They are often, but not always, numerical. • The classic notation is that denotes the f ( x ) output of a function at input value f x. • Functions are abstractions, but are very convenient for drawing mathematical relationships, and for analyzing these relationships.

Function or not? One of the key properties of a function is that it assigns a unique output to an input.

Vertical Line Test A trick for checking if a mathematical relationship plotted in the Cartesian plane is a function is the vertical line test. VLT: A plot is a function if and only if every vertical line intersects the plot in at most one place.

Sketch the following relationships in the Cartesian plane, and determine if they are functions. y = x + 1

x = | y |

x = y 3

2.2 Algebra of Functions

• Functions may be treated as algebraic objects: they may added, subtracted, multiplied, and divided in natural ways. • One must take care in dividing by functions that can be 0. Division by 0 is not defined. • There is one important operation of functions that does not apply to numbers: the operation of composition. • In essence, composing functions means applying one function, then the other.

Composition of Functions Given two functions , the composition of with is f ( x ) , g ( x ) f ( x ) g ( x ) ( f � g )( x ) denoted , and is defined as: ( f � g )( x ) = f ( g ( x )) . ( g � f )( x ) = g ( f ( x )) Similarly, . One thinks of as first applying the rule , then applying the ( f � g )( x ) f ( x ) g ( x ) rule .

As an example, consider . By f ( x ) = x + 1 , g ( x ) = x 2 substituting into , one sees that g ( x ) f ( x ) ( f � g )( x ) = x 2 + 1 f ( x ) g ( x ) Similarly, one can substitute into to compute that ( g � f )( x ) = ( x + 1) 2 = x 2 + 2 x + 1 In particular, we see that composition is not commutative , i.e. ( f � g )( x ) 6 = ( g � f )( x )

For the following pairs of functions, compute ( f + g )( x ) , ( fg )( x ) , ( f � g )( x ) f ( x ) = 3 x + 3 g ( x ) = x − 1

f ( x ) = sin( x ) g ( x ) = 2 x − 3

f ( x ) = x 2 + x − 2 g ( x ) = e x

2.3 Domain and Range of a Function

Let be a function. f ( x ) • The domain of is the set of allowable inputs. f ( x ) • The range of is the set of possible outputs f ( x ) for the function. • These can depend on the relationship the functions are modeling, or be intrinsic to the mathematical function itself. • They can also be inferred from the plot of , if f ( x ) it is available.

Intrinsic Domain Limitations Some mathematical objects have intrinsic limitations on their domains and ranges. Classic examples include: • has domain , range . f ( x ) = x 2 ( −∞ , ∞ ) [0 , ∞ ) • has domain , range . f ( x ) = √ x [0 , ∞ ) [0 , ∞ ) • has domain , range . f ( x ) = log( x ) ( −∞ , ∞ ) (0 , ∞ ) • has domain , range . f ( x ) = a x ( −∞ , ∞ ) (0 , ∞ ) f ( x ) = 1 • has domain and range . ( ∞ , 0) ∪ (0 , ∞ ) x

Visualizing Domain and Range Given a plot of , one can observe the domain f ( x ) and range by considering what and values x y are achieved. The function x f ( x ) = x 2 + 1 is hard to analyze, but its plot helps us guess its domain and range.

For the following functions, compute the domain and range and sketch a plot. √ f ( x ) = 1 − x

f ( x ) = 2 x + 1

f ( x ) = − ln(2 x + 1)

2.4 Inverse Functions

Let be a function. The inverse function f is f ( x ) the function that “undoes” ; it is f ( x ) denoted . f − 1 ( x ) More precisely, for all in the domain of , f ( x ) x ( f − 1 � f )( x ) = ( f � f − 1 )( x ) = x

Remarks on Inverse Functions • Not all functions have inverse functions; we will show how to check this shortly. • Note that , that is, inverse functions f − 1 ( x ) 6 = ( f ( x )) − 1 are not the same as the reciprocal of a function. The notation is subtle. • The domain of is the range of , and f − 1 ( x ) f ( x ) the range of is the domain of . f − 1 ( x ) f ( x ) • The plot of is the same as that of , f − 1 ( x ) f ( x ) except flipped over the line y = x.

Horizontal Line Test • Recall that one can check if a plot in the Cartesian plane is the plot of a function via the vertical line test. • One can check whether a function has an inverse function f ( x ) via the horizontal line test : the function has an inverse if every horizontal line intersects the plot of at most once. f ( x )

Computing Inverse Functions To compute an inverse function to , simply switch the role y = f ( x ) of the input and output variables and solve in terms of . x = f ( y ) x

For each of the following, plot the function and determine if it has an inverse function. If so, compute it. f ( x ) = x 2

f ( x ) = x 3

f ( x ) = e x

f ( x ) = cos( x )