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) or not Determine the domain for √x 2 - 4x + 3 (b)

2. 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 TEST If 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.

3. POLYNOMIAL FUNCTIONS: P(x) = a n x n + a n-1 x n-1 + a n-2 x n-2 +... 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)

4. 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 numbers Range: {5} Not onto Not 1-1

5. 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 numbers Range: All real numbers Onto One-to-One

6. 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 = ax 2 + bx + c y = x 2 + 2 Domain: All real numbers Range: [2, oo) from IR --> IR Not 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 + x 2 f(x) = - x 2 + 8x - 1 y = x 2 - 5x - 7

7. 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 numbers Range: [1,oo) Not onto Not 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