aim: how do we determine whether a function is one - to one, determine domain and range? do now:...

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Aim: How do we determine whether a function is one - to one, determine domain and range? Do Now: Determine whether the following function is onto (surjective) or not Determine the domain for √x 2 - 4x + 3 (b)

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POLYNOMIAL FUNCTIONS: P(x) = a n x n + a n-1 x n-1 + a n-2 x n a 1 x 1 + a 0 x 0 Ex amples: p(x) = 2 (polynomial of degree 0) p(x) = x 2 - 4x + 1 (polynomial of degree 2) p(x) = 7x 6 - 5x 2 + 9x - 3 (polynomial of degree 6)

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Page 1: Aim: How do we determine whether a function is one - to one, determine domain and range? Do Now: Determine whether the following function is onto (surjective)

Aim: How do we determine whether a function is one - to one, determine domain and range?Do Now: Determine whether the following function is onto (surjective) or not

Determine the domain for √x2 - 4x + 3(b)

Page 2: Aim: How do we determine whether a function is one - to one, determine domain and range? Do Now: Determine whether the following function is onto (surjective)

A function that maps values of set A to values in set B is one to one if and only if no two values in set A map to the same value in set B ( A one - to - one function is injective)

A function that is both injective ( one-to-one) and surjective (onto) is bijective

If a function is bijective then the domain has the same number of elements as the range

HORIZONTAL LINE TESTIf a horizontal line can be drawn such that it intersects the graph of a function more than once, then the function is NOT 1-1 .

Page 3: Aim: How do we determine whether a function is one - to one, determine domain and range? Do Now: Determine whether the following function is onto (surjective)

POLYNOMIAL FUNCTIONS:

P(x) = anxn + an-1xn-1 + an-2xn-2 + ... a1x1 + a0x0

Examples:

p(x) = 2 (polynomial of degree 0)p(x) = x2 - 4x + 1 (polynomial of degree 2)p(x) = 7x6 - 5x2 + 9x - 3 (polynomial of degree 6)

Page 4: Aim: How do we determine whether a function is one - to one, determine domain and range? Do Now: Determine whether the following function is onto (surjective)

For each function determine the domain, and range, and whether or not the function is onto, one-to-one, both or neither

Constant Function : y = a , a is a real number

y = 5

Domain: all real numbersRange: {5} Not ontoNot 1-1

Page 5: Aim: How do we determine whether a function is one - to one, determine domain and range? Do Now: Determine whether the following function is onto (surjective)

For each function determine the domain, and range, and whether or not the function is onto, one-to-one, both or neither

Linear Function: y = mx + b

y = -3x + 3

Domain: All real numbersRange: All real numbersOntoOne-to-One

Page 6: Aim: How do we determine whether a function is one - to one, determine domain and range? Do Now: Determine whether the following function is onto (surjective)

For each function determine the domain, and range, and whether or not the function is onto, one-to-one, both or neither

Quadratic function: y = ax2 + bx + c

y = x2 + 2Domain: All real numbersRange: [2, oo)

from IR --> IRNot onto Not one-to-one

To determine the range: Either find the turning point or use the range for p(x) = (x + b)2

Determine the domain and range for y = ( x- 6)2 + 3 h(x) = - 4 + x2

f(x) = - x2 + 8x - 1 y = x2 - 5x - 7

Page 7: Aim: How do we determine whether a function is one - to one, determine domain and range? Do Now: Determine whether the following function is onto (surjective)

For each function determine the domain, and range, and whether or not the function is onto, one-to-one, both or neither

absolute value function

y = | x - 2 | + 1

Domain: All real numbersRange: [1,oo)Not ontoNot 1-1

To determine the range realize that the range of f(x) = |ax + b| , a,b are real numbers is [0,oo)

Find the range of

Page 8: Aim: How do we determine whether a function is one - to one, determine domain and range? Do Now: Determine whether the following function is onto (surjective)