Chapter 1 Linear Functions and Mathematical Modeling Section 1.3

Section 1.3 Functions: Definition, Notation, and Evaluation Definition of Function Verbal, Numeric, Symbolic, and Graphical Descriptions of Functions Domain and Range Function Notation Evaluating Functions in Different Contexts

Definition of a Function A function is a relationship between two variables such that for each input there exists a unique output. one and only one ruleoutput y = 5x y = 15 input x = 3

Suppose input x = letter and output y = mailbox This is a function: Each input has one and only one output. This is not a function: There is an input with more than one output. (Cannot deliver letter x 3 to two different mailboxes!)

Example of a Function Verbal Description: The total salary for a math tutor will be calculated by multiplying the number of hours worked times the hourly rate. Numeric (Tabular) Description: Suppose the tutor earns $15 per hour. Number of hours worked, x Salary in dollars, y

Example of a Function (contd.) Symbolic Description: y = 15x where x represents number of hours worked, and y is the tutor’s total salary. Graphical Description:

True or False: The following table represents a function. True: For each input, there exists only one output Different inputs can share the same output, like (–3, 0) and (12, 0) p q07-20

Are these functions? x012 y64 -7 x012 y50512 x0494 y Yes; each input (or x-value) has one and only one output (or y-value). Yes; each input has one and only one output. (Reminder: x-values cannot repeat, but y-values can repeat.) No; input “4” is repeated. x = 4 does not have a unique (one and only one) y-value.

Domain and Range of a Function Domain: Possible values of the input. Range: Possible values of the output. Find the domain and range of the following function: Domain: {2004, 2007, 2010, 2011, 2013} Range: {85, 68, 124, 178, 205} Year, x Number of Visitors to a Museum (in thousands), y

Find the domain and range of the function: Domain: Since division by zero is undefined (it is mathematically impossible), the function is undefined for all the x-values that yield a 0 in the divisor (denominator). So, we must exclude any x-values that result in division by zero. Therefore, the domain is the set of all real numbers, with one restriction, x ≠ 6. In interval notation, (– , 6) U (6,  ). Range: The only way the given function would equal zero is to have a numerator 0, which is not the case. The range is the set of all real numbers except 0. In interval notation (– , 0) U (0,  ).

Find the domain and range of the function: Domain: The square root of a number is a real number only if the radicand is nonnegative. 3x – 12  0 3x  12 x  4 or [4,  ) Range: Taking the square root will result in a nonnegative number, thus the outputs of this function are 0 or positive. y  0 or [0,  )

Main Restrictions for Domain 1. Exclude any x-values that result in division by zero. 2. Exclude any x-values that result in even roots of negative numbers. That is, any x-values which make an expression under a square root (or any even root) negative.

Function Notation It is always useful to give a function a name; the most common name is “f.” We can also use other symbols or letters like g, h, p, etc. If x represents an input and y represents the corresponding output, the function notation is given by f(x) = y. That is: f(x) is the output for the function f when the input is x. f(input) = output. f(x) is read “f of x” or “the value of f at x.” Note: In function notation, f(x) does not mean multiplication of f times x.

Evaluating a Function If f(x) = 2x² – x + 3, find f(–5) f(–5) means to find the value of the function when the input variable has a value of –5. Substitute –5 for x and simplify. f(–5) = 2(–5)² – (–5) + 3 = 2(25) = = 58 f(–5) = 58; that is, when the input is –5, the output is 58.

If f(x) = x² – 3x + 4, find f( a ). Substitute “ a ” for x and simplify. f( a ) = ( a )² – 3( a ) + 4 = a ² – 3 a + 4 Since we have no numerical value for a, we stop!

If f(x) = x² – 3x, find f( a + 2). Substitute “ a + 2” for x and simplify. f( a + 2) = ( a + 2)² – 3( a + 2) = ( a + 2)( a + 2) – 3( a + 2) = a ² + 4 a + 4 – 3 a – 6 = a ² + a – 2 Caution: ( a + 2)² ≠ a ² + 4

Dance Rooms Here charges $65 per hour and a $350 deposit. The table below illustrates the total cost (in dollars) of renting a dance room for different number of hours. a. Find the value(s) of x when f(x) = 675. We want to find the input when the output is 675. Therefore, x = 5. b. Find and interpret f(4.5). We want to find the output when the input is 4.5. Therefore, f(x) = If the dance room is used 4.5 hours, the total cost is $ Hours, x Total cost, f(x)

Using your textbook, practice the problems assigned by your instructor to review the concepts from Section 1.3.