Chapter 1 – Functions and Their Graphs

Functions Section 1

Introduction to Functions Definition – A function f from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set A is the domain of the function f, and the set B contains the range

Characteristics of a Function Each element in A must be matched with an element in B. Some elements in B may not be matched with any element in A. Two or more elements in A may be matched with the same element in B. An element in A (domain) cannot be match with two different elements in B

Example A = {1,2,3,4,5,6} and B = {9,10,12,13,15} Is the set of ordered pairs a function? {(1,9), (2,13), (3,15), (4,15), (5,12), (6,10)}

Vertical Line Test Use the vertical line test to determine graphically when you have a function. If you can draw a vertical line and it does not pass through more than one point on the graph, then the graph depicts a function.

Function Notation The variable f is usually used to depict a function. It is only notation, and f(x) simply replaces y in your typical equations and is read f of x. Therefore y = f(x) That means if y = 2x +4 then an equivalent equation using function notation is f(x) = 2x + 4 Nothing changes, it’s just another use of symbols.

Example Evaluate the function when x = -1, 0, and 1 f(x) = { x2 +1, x< 0 { x -1, x ≥ 0 f(-1) = (-1)2 +1 = 2 f(0) = 0 -1 = -1 f(1) = 1 – 1) = 0 f(x) = 1 – x2 then f(1) = 1 – (1) 2 = 0 f(2) = 1 – (2) 2 = -3 f(0) = 1 – (0) 2 = 1

Domain of a Function The domain of a function is the set of all real numbers for which the expression is defined. EXAMPLE f(x) = 1/(x2 -4) The domain is the set of real numbers excluding ± 2.

Analyzing Graphs of Functions Section 2

Graph of a Function The graph of a function f is the collection of ordered pairs (x,f(x)) such that x is in the domain of f. Domain – is the set of all x values Range – is the set of all f(x) values

Zeros of a Function The zeros of a function f of x are the x-values for which f(x) = 0 EXAMPLE Find the zeros of f(x) = 3x2 +x - 10 3x2 +x – 10 = 0 - Factor and solve for x

Increasing and Decreasing Functions A function f is increasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2) A function f is decreasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2) A function f is constant on an interval if, for any x1 and x2 in the interval, f(x1) = f(x2)

Example Graph f(x) = x3 Graph f(x) = x3 – 3x

Definition of Relative Minimum and Relative Maximum A function value f(a) is called a relative minimum of f if there exist an interval (x1, x2) that contains a such that x1 < x < x2 implies f(a) ≤ f(x) A function value f(a) is called a relative maximum of f if there exist an interval (x1, x2) that contains a such that x1 < x < x2 implies f(a) ≥ f(x)

Example Graph f(x) = 3x2 – 4x -2 using a calculator to estimate the relative minimum or relative maximum Graph f(x) = -3x2 + 4x + 2 using a calculator to estimate the relative minimum or relative maximum

Types of Functions Linear Functions: Step Functions: f(x) = mx + b Step Functions: f(x) = [[ x ]] = greatest integer less than or equal to x Piecewise-Defined Functions: f(x) = {2x +3, x ≤ 1 {- x + 4, x > 1

Even and Odd Functions A function y = f(x) is even if, for each x in the domain of f, f(-x) = f(x) --- symmetric to y-axis A function y = f(x) is odd if, for each x in the domain of f, f(-x) = - f(x) --- symmetric to origin

Example Determine whether each function is even, odd, or neither g(x) = x3 –x h(x) = x2 + 1

Shifting, Reflecting, and Stretching Graphs Section 3

Summary of Graphs of Common Functions f(x) = c f(x) = x f(x) = |x| f(x) =  x f(x) = x2 f(x) = x3

EXAMPLE Shifting Graphs Transforms graphs by shifting upward, downward, left or right with basic graph the same. EXAMPLE h(x) = x2 + 2 shifts the graph upward two units

Vertical Shifts h(x) = f(x) + c for c > 0 Vertical shift c units upward f(x) = f(x) – c for c > 0 Vertical shift c units downward

Horizontal Shifts h(x) = f(x – c) for c > 0 horizontal shift c units right f(x) = f(x + c) for c > 0 horizontal shift c units left

Reflecting in the Coordinate Axes Reflections in the x-axis: h(x) = - f(x) Reflections in the y-axis: h(x) = f(-x)

EXAMPLE Reflecting Graphs Transforms graphs by creating a mirror image If h(x) = x2 then g(x) = - x2 is the reflection

Nonrigid Transformation Transformations that cause a distortion – a change in the shape of the original graph If h(x) = |x| then g(x) = 3|x| is a vertical stretch of h(x) but p(x) = ⅓|x| would be a vertical shrink

Combinations of Functions Section 4

Arithmetic Combinations of Functions (f +g)(x) = f(x) + g(x) sum (f -g)(x) = f(x) - g(x) difference (fg)(x) = f(x) · g(x) product (f/g)(x) = f(x)/g(x), g(x) ≠ 0 quotient

Examples f(x) = 2x + 1 and g(x) = x2 + 2x – 1 Find: (f +g)(x) = f(x) + g(x) = x2 + 4x Find: (fg)(x) = f(x)· g(x) =2 x3 +5x2 - 1

Composition of Functions The composition of the function f with the function g is (f ◦ g)(x) = f(g(x)) The domain of f ◦ g is the set of all x in the domain of g such that g(x) is in the domain of f

Examples f(x) = x + 2 and g(x) = 4 – x2 Find: (f ◦ g)(x) = f(g(x)) = f(4 – x2) Simplify = 4 – x2+2 Find: (g ◦ f)(x) = g(f(x)) = g(x + 2) Simplify = 4 – (x +2)2

Examples f(x) = x2 - 9 and g(x) = (9 - x2)½ Find: domain of (f ◦ g) Remember the domain of (f ◦ g) is the set of all x in the domain of g Find domain of g(x):

Inverse of Functions Section 5

Inverse Functions Let f and g be two functions such that f(g(x)) = x for every x in the domain of g and; g(f(x)) = x for every x in the domain of f Under these conditions, the function g is the inverse function of the function f

Inverse Functions The inverse function is formed by interchanging the first and second coordinates of each of the ordered pairs and the inverse is denoted by f -1 Again, this is simply notation! The domain of f must be equal to the range of f -1 , and the range of f must be equal to the domain of f-1

Example Find the inverse function of f(x) = 2x - 3 Replace f(x) with y and solve for x y = 2x -3 x = (y+3)/2 Now interchange x and y and you have f -1 y = (x+3)/2

Guidelines for Finding an Inverse Function Use the Horizontal Line Test to decide whether f has an inverse function In the equation for f(x), replace f(x) by y Interchange the roles of x and y, and solve for y. Replace y by f-1(x) in the new equation Verify that f and f-1 are inverse functions of each other by showing that the domain of f is equal to the range of f -1 and the range of f is equal to the domain of f -1

Mathematical Modeling Section 6

Direct Variation The following statements are equivalent. y varies directly as x. y is directly proportional to x y = kx for some nonzero constant k EXAMPLE D = rt F= ma

Inverse Variation The following statements are equivalent. y varies inversely as x. y is inversely proportional to x y = k/x for some nonzero constant k EXAMPLE V = kT/P

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