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1.5 Inverse Functions. Properties, Domain and Range. Goal. You should be able to determine the inverse of linear functions and state their properties. Inverse Functions. The reverse of the original function Maps each output value back to the corresponding input value
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1.5 Inverse Functions Properties, Domain and Range
Goal • You should be able to determine the inverse of linear functions and state their properties.
Inverse Functions • The reverse of the original function • Maps each output value back to the corresponding input value • The "undo" of a function
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
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
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)
TOV f(x) f-1(x)
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
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
Class/Home work • Pg 46 #1a, 2 – 4, 6, 7, 10, 12, 16, 17