Download - What is a Function? by Judy Ahrens ~ 2005, 2006 Pellissippi State Technical Community College
What is aFunction?
by Judy Ahrens ~ 2005, 2006Pellissippi State Technical
Community College
Definition of a Function
slide 1
2x 3 13x = 4
3x2
6x = 4
4x 5 21x = 4
x x –2x = 4
REMEMBER!
Ima
Function
A function is a special relationship between twosets of elements. When you choose one elementfrom the first set, there must be exactly oneelement in the second set which goes with it.
ONE IN
ONE OUT
The Vertical Line TestIt 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 a function
a function
not a function
not a functionnot a function
Recognizing a Function
slide 3
–2 0 3 4
4 0 916
The first relationship below defines a function,
01
4
0 1–1 2
but the second does not. Why not??
0149
1
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 input
one 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, then2x2 36x 3xIf f(x) = , then
f(0) = 4(0) + 5 = 52
f(-7) = 3(-7) – 6 = 141
2 36( ) 3( ) 2 4 02 2 4 2 0 f(2) =
Domain of a Function
slide 5
–2 0 3 4
4 0 916
The domains below are sets of individual numbers.
D: {–2,0,3,4}
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1
D: {0,1,4,9}
The domains of the functions below are intervals.
D: ( ,2]
{(–5,2), (0,1), (4,–9), (7,6)}
D: { 5,0,4,7}
D: ( , )
The set of all x-values (inputs) is the domain.
D: (0, )
Implied DomainPolynomial Functions
slide 6
The domain of all polynomial functions is the set of all real numbers, i.e. . Examples:( , )
f (x) 3
f (x) 3x 8
Constant function
Linear function
Quadratic function
Cubic function
2 4x 1g(x) x 3 2h(x) x 8x 5
A term is a product of a number and a nonnegative integer power of a variable, e.g. 4 32x , 3x, 7, xA polynomial function is the sum or difference of terms, e.g. 4 3f (x) 2x 3x 7 x
Implied DomainRational Functions
slide 7
The domain of a rational function is the set ofall real numbers except those which would make the denominator equal zero.
3xf (x)2x 8
2
7g(x)x x 6
Examples:
2x 8 0 x 4 2x x 6 0
A rational function is the quotient of two polynomial functions, e.g.
3xf (x)2x 8
D : {x | x 4}, i.e.( , 4) (4, )
(x 3)(x 2) 0 x 2, 3 D : {x | x 2, 3}, i.e.( , 2) ( 2, 3) (3, )
Implied DomainRadical Functions
slide 8
The domain of radical functions with odd indicesis the set of all real numbers, i.e. .( , )
Examples: 3f (x) 3x 9,
The domain of a radical function with even indexis the set of all real numbers except those which make the radicand negative. .
Examples:
3x 9 0 x 3
f (x) 3x 9 4g(x) 8 4x
D : {x | x 2},i,e, [3, )D : {x | x 3},
8 4x 0 x 2
i.e. ( , 2]
5g(x) 8 4x
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: [0, )
R: { 9, 1 ,2}
R: ( , )
The set of all y-values (outputs) is the range.
R: ( , 7]
–2 0 3 4
4 0 916
0149
1{(–5,2), (0,1), (4,–9), (7,1)}
For Practice
slide 10
1. 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: 4f (x) 3x 2x 1
4. Find the domain of each function: 3 2g(x) 6x 2, f (x) 4 2x,
5xg(x)
3x 6
5. Find D & R
1. No, 4 is paired with 2 different numbers 2. 20, 3. Both are ( , ) 4.
( , ), ( ,2], ( , 2) ( 2, )
6. Are these graphs of functions?
D: ( , 2], R: ( ,5] 5.
The End
f(0), f(b), f(2a-b)
8-6a+3b8, 8-3b,
yes6. No, it fails the VLT;