1.3 Functions

Domain and RangeFunction notationPiecewise functionsInterval NotationDifference quotient

Domain and Range

Domain is the Input (Independent variable) used in a function.

Range is the Output (Dependent variable) given by the function.

Relations are functions, functions match elements from the domain to the range

A Function

A car’s brake pedal is part of a function car.

Step on the brake, the car should stop.

Every time.

Elements in the Domain only map to one element in the Range

a x An element in the

b y Range can have

c z many Inputs, Where

d w the Domain can only

go to one element in the Range.

Ways to Represent a function

Verbally How the input effect the output

Numerically A table of numbers

Graphically With a graph

AlgebraicallyWith an equation

Testing a Function

Verbally with a Proof

Numerically with a table. If an element from the Domain has two different outcomes, it is not a function. For example (1, 5), (2, 3) and (2,1).

Testing a function

Graphically uses the vertical line test. If a vertical line touches the graph in more then one spot, then the graph is not the graph of a function.

Algebraically is where you solve an element of the range. X2+ y= 8 Solve for y= - x2+8

Y has only one answer so it is ok

x2+y2=25 Solve for Not a function

Function notation

Use f(x) to stand in for y in the equation

y = 5x – 2 , so it becomes f(x) = 5x – 2

Why would we need this notation?

Solve a function with different inputs

Let k(x) = 2x2-1

So find k(x) for x ={0, 1, 2}

k(0) = 2(0)2- 1 = -1

k(1) = 2(1)2- 1 = 1

k(2) = 2(2)2- 1 = 7

Sometime the input can be an expression

h(x) = x2 – 4x + 6

Let x = 2

h(2) = (2)2 – 4(2) + 6 = 2

Let x = (x + 1)

h(x+1) = (x+1)2 – 4(x+1) + 6

h(x+1) = (x2 2x + 1) – 4x – 4 + 6

h(x+1) = x2 – 2x + 3

Piecewise functionsdifferent functions over different parts of a domain

Here is the Rule of the piecewise function

(0,2)

If x = - 2 , then 2(-2) + 2 = -2

If x = 0 , then -3(0) – 1 = -1

If x = 1, then -3(2) – 1 = -4 (0, - 1) (1, -4)

(-2, -2)

Implied Domain, the real numbers in which the function is defined

Domain where x does not equal 5 or All real numbers except 5

All real numbers except

Where x greater then or equal to 0

Ways to find the Implied DomainsFind what makes the denominator of a fraction

zero.

Find what makes an even root (square root, 4th roots and so on) inside negative.

Interval Notation

Find the Implied DomainAfter setting 9 – x2 = 0 and finding the answer -3 ≤ x ≤ 3 , easier said then done.Since x includes -3 and 3 we use [ ] to show the number betweenSo -3 ≤ x ≤ 3 , becomes [-3, 3]If it was 4 < y ≤ 10, it would be (4, 10]( ) shows that the end numbers are not included in the solution set.

The Difference Quotient

A

f(x) = x2+ 3x + 5

In the Difference Quotient

(x+h)2+3(x+h)+5-(x2+3x+5) h

= x2+2hx+h2+3x+3h+5-x2-3x-5

h

= 2hx + h2+3h = h(2x + h + 3)

h h

= 2x + h + 3

h not equaling 0

Homework

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Homework

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