Sections 3.1 - 3.3 Dr. Doug Ensley January 13, 2015
Function Machine ◮ A function accepts an input x and returns an output f ( x ). ◮ The set of all legitimate inputs is called the domain , and the set of all outputs that occur is called the range . ◮ A function can associate only one output to any given input. It is ok for a function to have the same output for more than one input.
Function Example ◮ The function with the rule, “For each input number x , return as output the number x 2 + 1,” can be written more concisely as, f ( x ) = x 2 + 1. When we can provide such an algebraic rule, then the definition of function is usually met.
Function Example ◮ The function with the rule, “For each input number x , return as output the number x 2 + 1,” can be written more concisely as, f ( x ) = x 2 + 1. When we can provide such an algebraic rule, then the definition of function is usually met. ◮ The function f ( x ) = x 2 + 1 has as its domain the set of all real numbers, denoted R , or in interval notation ( −∞ , ∞ ).
Function Example ◮ The function with the rule, “For each input number x , return as output the number x 2 + 1,” can be written more concisely as, f ( x ) = x 2 + 1. When we can provide such an algebraic rule, then the definition of function is usually met. ◮ The function f ( x ) = x 2 + 1 has as its domain the set of all real numbers, denoted R , or in interval notation ( −∞ , ∞ ). ◮ The graph of a function consists of all points ( x , y ) where y is the output associated with input x . I will refer to such points as “input, output pairs,” and we often use the notation y = f ( x ) when referring to a graph of a function.
Function Evaluation Let f ( x ) = x 2 − 2 x + 3. Evaluate each of the following: ◮ f ( − 3)
Function Evaluation Let f ( x ) = x 2 − 2 x + 3. Evaluate each of the following: ◮ f ( − 3) � 1 ◮ f � 2
Function Evaluation Let f ( x ) = x 2 − 2 x + 3. Evaluate each of the following: ◮ f ( − 3) � 1 ◮ f � 2 ◮ f ( − x ) (and simplify)
Function Evaluation Let f ( x ) = x 2 − 2 x + 3. Evaluate each of the following: ◮ f ( − 3) � 1 ◮ f � 2 ◮ f ( − x ) (and simplify) ◮ f ( x + h ) (and simplify)
Function Evaluation Let f ( x ) = x 2 − 2 x + 3. Evaluate each of the following: ◮ f ( − 3) � 1 ◮ f � 2 ◮ f ( − x ) (and simplify) ◮ f ( x + h ) (and simplify) ◮ f ( x + h ) − f ( x ) (and simplify) h
Function Evaluation Let f ( x ) = x 2 + 1. Evaluate each of the following: ◮ f (7)
Function Evaluation Let f ( x ) = x 2 + 1. Evaluate each of the following: ◮ f (7) � 1 ◮ f � 3
Function Evaluation Let f ( x ) = x 2 + 1. Evaluate each of the following: ◮ f (7) � 1 ◮ f � 3 ◮ f (2 x ) (and simplify)
Function Evaluation Let f ( x ) = x 2 + 1. Evaluate each of the following: ◮ f (7) � 1 ◮ f � 3 ◮ f (2 x ) (and simplify) ◮ f ( x + h ) (and simplify)
Function Evaluation Let f ( x ) = x 2 + 1. Evaluate each of the following: ◮ f (7) � 1 ◮ f � 3 ◮ f (2 x ) (and simplify) ◮ f ( x + h ) (and simplify) ◮ f ( x + h ) − f ( x ) (and simplify) h
More Function Examples ◮ The function with the rule, “For each input number x , return x +1 as output the number √ x − 3 ,” can be written more concisely as, x + 1 g ( x ) = √ x − 3
More Function Examples ◮ The function with the rule, “For each input number x , return x +1 as output the number √ x − 3 ,” can be written more concisely as, x + 1 g ( x ) = √ x − 3 ◮ The function g ( x ) has as its domain the set of all real numbers greater than 3.
More Function Examples ◮ The function with the rule, “For each input number x , return x +1 as output the number √ x − 3 ,” can be written more concisely as, x + 1 g ( x ) = √ x − 3 ◮ The function g ( x ) has as its domain the set of all real numbers greater than 3. ◮ In interval notation, the domain is written (3 , ∞ ).
Domain ◮ What is the domain of the function g ( x ) = √ 2 x − 5?
Domain ◮ What is the domain of the function g ( x ) = √ 2 x − 5? ◮ What is the domain of the function g ( x ) = x +1 x − 4 ?
Domain ◮ What is the domain of the function g ( x ) = √ 2 x − 5? ◮ What is the domain of the function g ( x ) = x +1 x − 4 ? ◮ What is the domain of the function g ( x ) = x 2 + x +1 √ 3 x − 6 ?
Domain ◮ What is the domain of the function g ( x ) = √ 2 x − 5? ◮ What is the domain of the function g ( x ) = x +1 x − 4 ? ◮ What is the domain of the function g ( x ) = x 2 + x +1 √ 3 x − 6 ? 1 ◮ What is the domain of the function g ( x ) = x 2 +5 ?
Functions from Word Problems ◮ The function with the rule, “For each input number x , return as output the price in dollars of purchasing x candy bars that cost $0.79 each,” can be written algebraically as, P ( x ) = 0 . 79 · x . What is the domain of the function P ( x )?
Functions from Word Problems ◮ The function with the rule, “For each input number x , return as output the price in dollars of purchasing x candy bars that cost $0.79 each,” can be written algebraically as, P ( x ) = 0 . 79 · x . What is the domain of the function P ( x )? ◮ A rectangular garden has a perimeter of 100 feet. Find an algebraic expression for the area A ( w ) of the garden if the width is the input number w . What is the domain of the function A ( w )?
Functions from Word Problems ◮ The function with the rule, “For each input number x , return as output the price in dollars of purchasing x candy bars that cost $0.79 each,” can be written algebraically as, P ( x ) = 0 . 79 · x . What is the domain of the function P ( x )? ◮ A rectangular garden has a perimeter of 100 feet. Find an algebraic expression for the area A ( w ) of the garden if the width is the input number w . What is the domain of the function A ( w )? ◮ A popular burger restaurant gets a revenue R , in dollars, from the sale of x hundred burgers equal to R ( x ) = − 2 . 4 x 2 + 440 x . The cost C , in dollars, of selling x hundred burgers is given by the function C ( x ) = 0 . 1 x 3 − 4 x 2 + 130 x + 1000. Find the profit if 7 hundred burgers are sold. If P ( x ) denotes the profit when selling x hundred burgers, find P (7) and write a sentence explaining its meaning.
Graph of a function ◮ The graph of a function consists of all points ( x , y ) where y is the output associated with input x . I will refer to such points as “input, output pairs,” and we often use the notation y = f ( x ) when referring to a graph of a function.
Graph of a function ◮ The graph of a function consists of all points ( x , y ) where y is the output associated with input x . I will refer to such points as “input, output pairs,” and we often use the notation y = f ( x ) when referring to a graph of a function. 4 x 2 + 1 is shown below. ◮ For example, the graph of f ( x ) = 1
Graph of a function ◮ The graph of a function consists of all points ( x , y ) where y is the output associated with input x . I will refer to such points as “input, output pairs,” and we often use the notation y = f ( x ) when referring to a graph of a function. 4 x 2 + 1 is shown below. ◮ For example, the graph of f ( x ) = 1 ◮ When we want to refer to a point on the graph, we can specify just the x coordinate or we can describe both coordinates as, for example, ( − 2 , f ( − 2)).
Graph of a function Consider the graph y = f ( x ) shown below: ◮ Compute f (3) and f (6).
Graph of a function Consider the graph y = f ( x ) shown below: ◮ Compute f (3) and f (6). ◮ The point (8 , a ) is on the graph. What is a ? The point ( b , − 2) is on the graph. What is b ?
Graph of a function Consider the graph y = f ( x ) shown below: ◮ Compute f (3) and f (6). ◮ The point (8 , a ) is on the graph. What is a ? The point ( b , − 2) is on the graph. What is b ? ◮ How many solutions does the equation f(x)=1 have?
Graph of a function Consider the graph y = f ( x ) shown below: ◮ Compute f (3) and f (6). ◮ The point (8 , a ) is on the graph. What is a ? The point ( b , − 2) is on the graph. What is b ? ◮ How many solutions does the equation f(x)=1 have? ◮ Identify the x -intercepts for this function?
Graph of a function Consider the graph y = f ( x ) shown below: ◮ Compute f (3) and f (6). ◮ The point (8 , a ) is on the graph. What is a ? The point ( b , − 2) is on the graph. What is b ? ◮ How many solutions does the equation f(x)=1 have? ◮ Identify the x -intercepts for this function? ◮ Identify the y -intercepts for this function?
Graph of a function Consider the graph y = f ( x ) shown below: ◮ Compute f (3) and f (6). ◮ The point (8 , a ) is on the graph. What is a ? The point ( b , − 2) is on the graph. What is b ? ◮ How many solutions does the equation f(x)=1 have? ◮ Identify the x -intercepts for this function? ◮ Identify the y -intercepts for this function? ◮ What is the domain of this function? What is the range?
Increasing/Decreasing Consider the following graph of y = f ( x ). ◮ Where is f increasing? ◮ Where is f decreasing? ◮ Where are the “turning points?”
Local extrema Where are the What are the Are there local maxima? local maxima? global extrema?
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