What is a Function?

1. What is aFunction? by Judy Ahrens ~ 2005, 2006 Pellissippi State Technical Community College

2. 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. Ima Function ONE IN ONE OUT x = 4 x = 4 21 13 x = 4 x = 4 6 –2 Definition of a Function REMEMBER! slide 1

3. not a function not a function not a function 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. a function a function a function slide 2

4. The first relationship below defines a function, but the second does not. Why not?? What about the third relationship? 0 1 –1 2 0 1 4 9 0 1 4 4 0 9 16 –2 0 3 4 1 Recognizing a Function Remember! One input one output 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! slide 3

5. 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 f(-7) = 3(-7) – 6 = 141 If f(x) = , then f(2) = Functional Notation Equations are frequently used to represent functions. “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(0) = 4(0) + 5 = 5 slide 4

6. {(–5,2), (0,1), (4,–9), (7,6)} 0 1 4 9 4 0 9 16 –2 0 3 4 1 D: D: D: D: Domain of a Function The set of all x-values (inputs) is the domain. 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. slide 5

7. 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. The domain of all polynomial functions is the set of all real numbers, i.e. . Examples: Implied Domain Polynomial Functions Constant function Linear function Quadratic function Cubic function slide 6

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

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

10. The set of all y-values (outputs) is the range. {(–5,2), (0,1), (4,–9), (7,1)} 0 1 4 9 4 0 9 16 –2 0 3 4 1 R: R: R: R: Range of a Function These ranges are sets of individual numbers. R: {0, 4, 9, 16} R: {1} The ranges of these functions are intervals. slide 9

11. 3. Find the domain and range: 3. Both are 4. 5. 6. No, it fails the VLT; For Practice 1. Is it a function? {(0,0), (1,1), (4, 2), (4,-2)} 2. If f(x) = 8 – 3x: find f(-4), f(0), f(b), f(2a-b) 4. Find the domain of each function: 5. Find D & R 6. Are these graphs of functions? 1. No, 4 is paired with 2 different numbers 2. 20, 8, 8-3b, 8-6a+3b yes The End slide 10