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Inverse Functions

Learn about inverse functions and how to find them. Understand the concept of switching x and y and solving for y to find the inverse function. Explore examples with linear and power functions and see how to verify the inverse. Discover how the graph of the inverse is the reflection of the original function.

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Inverse Functions

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  1. Inverse Functions Winnie Chen, Gary Choi, Kayla Glufling, Jacky Chen

  2. What is an “inverse function”? A function that performs the REVERSE of the original function. Therefore, when the inverse is plugged in as X in the original equation, the answer would be y=x (vice versa). ƒ(g(x)) = x AND g(ƒ(x)) = x The function g would be denoted as ƒ-1 and read as “ƒ inverse”.

  3. How to find an inverse function Write the original relation y = 2 x — 4 Switch x and yx = 2 y — 4 Add 4 to both sides x + 4 = 2 y Dive both sides by 2 ½ x + 2 = y • The inverse relation of y = 2 x – 4 is y = ½ x + 4 With any given function, you can find its inverse by switching the places of x and y, then simply solve for y.

  4. How to verify an inverse function Verify that ƒ(x) = 2x—4 and ƒ-1 (x) = ½x+2 Using ƒ(ƒ-1(x)) = x Plug in the inverse into ƒ-1ƒ(ƒ-1(x)) = ƒ(½x+2) Plug in the original ƒ(ƒ-1(x)) = 2(½x+2)—4 Simplify ƒ(ƒ-1(x)) = x + 4 — 4 Solve ƒ(ƒ-1(x)) = x Using ƒ-1(ƒ(x)) = x Plug in the original into ƒ ƒ-1(ƒ(x)) = ƒ-1(2x—4) Plug in the inverse ƒ-1(ƒ(x)) = ½(2x—4) +2 Simplify ƒ-1(ƒ(x)) = x—2 +2 Solve ƒ-1(ƒ(x)) = x

  5. Input/output relation • The DOMAIN of the inverse relation is the RANGE of the original relation. • The RANGE of the inverse relation is the DOMAIN of the original relation.

  6. So, what does the graph look like? The graph of the inverse relation is simply the reflection of graph of the original relation. Therefore the line of reflection would be y = x **You can find the inverse relationby using the graph. Just switchthe range and domain of the original equation. Original Line of Symmetry Inverse

  7. How to find inverse of power functions Write the original relation: f(x)= 1/16x5 Switch x and y: x= 1/16y5 Multiply both sides by 16: 16*x = y5 Take both sides to the 1/5 power: (16x)1/5 = (y5)1/5 Simplify: (16x)1/5 = y Solve: y = 0.2x1/5

  8. How to find the inverse of a cubic function • Write the original function: f(x) = x3+4 • Substitute y into f(x): y = x3+4 • Switch x and y: x = y3+4 • Minus 4 on both sides: x – 4 = y3 • Cube root both sides:3√(x-4) = y • Substitute f-1(x) for y: f-1(x) = 3√(x-4)

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