What is a Function? by Judy Ahrens ~ 2005, 2006 Pellissippi State Technical Community College

Definition of a Function slide 1 13 x = x = 4 –2 x = 4 REMEMBER! Ima Function A function is a special relationship between two sets of elements. When you choose one element from the first set, there must be exactly one element in the second set which goes with it. ONE INONE OUT

The Vertical Line Test It is easy to recognize a function from its graph. A graph represents a function if and only if no vertical line intersects the graph more than once. slide 2 a function not a function

Recognizing a Function slide 3 – The first relationship below defines a function, –1 2 but the second does not. Why not?? What about the third relationship? The second relationship pairs “1” with both “1” and “–1”, so it is not a function. The third relationship defines a function; each first element is paired with exactly one second element. The second elements can be the same! Remember! One inputone output

Functional Notation Equations are frequently used to represent functions. slide 4 “x” is the independent variable “f(x)” is the dependent variable We may let y = f(x) on the graph of a function. If f(x) = 4x + 5, then f(x) = 4x + 5 is written in functional notation. We read it as “f of x equals 4x plus 5”. If f(x) = 3 – 6, then If f(x) =, then f(0) = 4(0) + 5 = 5 f(-7) = 3(-7) – 6 = 141 f(2) =

Domain of a Function slide 5 – The domains below are sets of individual numbers. D: {–2,0,3,4} D: {0,1,4,9} The domains of the functions below are intervals. D: {(–5,2), (0,1), (4,–9), (7,6)} D: The set of all x-values (inputs) is the domain. D:

Implied Domain Polynomial Functions slide 6 The domain of all polynomial functions is the set of all real numbers, i.e.. Examples: Constant function Linear function Quadratic function Cubic function A term is a product of a number and a nonnegative integer power of a variable, e.g. A polynomial function is the sum or difference of terms, e.g.

Implied Domain Rational Functions slide 7 The domain of a rational function is the set of all real numbers except those which would make the denominator equal zero. Examples: A rational function is the quotient of two polynomial functions, e.g.

Implied Domain Radical Functions slide 8 The domain of radical functions with odd indices is the set of all real numbers, i.e.. Examples: The domain of a radical function with even index is the set of all real numbers except those which make the radicand negative.. Examples:

Range of a Function slide 9 These ranges are sets of individual numbers. R: {0, 4, 9, 16}R: {1} The ranges of these functions are intervals. R: The set of all y-values (outputs) is the range. R: – {(–5,2), (0,1), (4,–9), (7,1)}

For Practice slide Is it a function? {(0,0), (1,1), (4, 2), (4,-2)} 2. If f(x) = 8 – 3x: find f(-4), 3. Find the domain and range: 4. Find the domain of each function: 5. Find D & R 1. No, 4 is paired with 2 different numbers 2. 20, 3. Both are4. 6. Are these graphs of functions? 5. The End f(0),f(b),f(2a-b) 8-6a+3b8,8-3b, yes 6. No, it fails the VLT;