1.5 Inverse Functions

1. 1.5 Inverse Functions Properties, Domain and Range

2. Goal • You should be able to determine the inverse of linear functions and state their properties.

3. Inverse Functions • The reverse of the original function • Maps each output value back to the corresponding input value • The "undo" of a function

4. Inverse Functions • The inverse of a linear function is the reverse of the original function. • It can be found by performing the inverse operations (division instead of multiplication, say) in reverse order. f(x) = 2x – 3 f-1(x) = x + 3 2

5. Inverse Functions • It can be found by exchanging the x and y variables in the expression and solving for y: y = 5x + 8 (exchange x and y) x = 5y + 8 y = x – 8 5

6. Inverse Functions • The inverse of a function is not necessarily a function itself. • The inverse function of f(x) is written as f-1(x). • If (a, b) is a point on the function y = f(x) then (b, a) is a point on y = f-1(x). • Note: f-1(x) ≠ 1/f(x)

8. TOV f(x) f-1(x)

9. Mapping Diagrams

10. Graphing Inverse Functions • When graphing an inverse of a function, you reflect it across the y = x line (as (x, y) of the function is (y, x) of the inverse). • Any points shared by both functions lie on y = x and are called invariant points (shared by both graphs) Complete 1.5 notes worksheet

11. Domain and Range • Given the following function determine the domain and range of both the function and its inverse. • D = {xЄR}, R = {f(x)ЄR} • Inverse: x = -2(y+3)2 – 8 • y = We know roots can’t be negative: when? • D = {xЄR|x ≤ – 8} and R = {f-1(x)ЄR| f-1(x) ≥ – 3} • Note: when x = – 8, f-1(– 8) = – 3, f-1(– 10) = – 2

12. Class/Home work • Pg 46 #1a, 2 – 4, 6, 7, 10, 12, 16, 17