Radical Functions and Equations ( 무리함수 ) Radical sign index The term is called a radical. The symbol is called a radical sign(sometimes just called radical or rad). The root number is called the index. The expression inside the radical is called the radicand.

A radical function is a function that has a variable in the radicand. Horizontal shift left one Vertical stretch two. Vertical shift down three You can apply the same transformations to the graphs of radical functions as you can to polynomial functions.

Radical Parent functions As x , f(x)  As x  0 +, f(x)  0 As x , f(x)  As x  – , f(x)  –  As x , f(x)  As x  0 +, f(x)  0

Transformations Vertical stretch of 3 Vertical shift up 2 Horizontal shift right 3 We apply the transformations on these functions in the same manner as we did with the polynomial functions. Domain: Range Domain: Range: Domain: Range:

Applying the transformations Vertical shrink of ½ Vertical stretch 3 Horizontal shift right 2 Vertical shift up 4 Horizontal shift left 1 Vertical shift down 1 Domain:Range: As x x , f(x)  ;; As x x -1 +, f(x)  As x x , f(x)  - ;; As x x 2 +, f(x)  4 Domain:Range: Reflect across x-axis

Applying the transformations Vertical stretch of 3 Horizontal shift left 2 Vertical shift down 4 Vertical shrink of ½ Horizontal shift right 1 Vertical shift up 2 Domain:Range: As x x , f(x)   ; As x x - , f(x)  -  Domain:Range: As x x , f(x)  -  ; As x x - , f(x)   x-axis

To solve a radical equation that has only one variable in the radicand, isolate that term on one side of the equation. If the index is 2, then square both sides of the equation. Be careful! The new equation you created when you Squared both sides might have extraneous solutions! Isolate the radical. Square both sides. Simplify and set equal to zero. Factor. Solve. Radical Equations – not on final exam Given:

These solutions may not be solutions to the original equation. Check your solutions ! The graph below illustrates that the derived equation may be different from the original equation. √ Solving Radical Equations

Try: Check: Try: Check

A radical equation may contain two radical expressions with an index of 2. To solve these, rewrite the equation with one of the radicals isolated on one side of the equals sign. Then, square both sides. If a variable remains in a radicand, you must repeat the squaring process. More Solving Radical Equations

Check: Try: More Solving Radical Equations Isolate one radical on each side of equals sign. Square both sides. Collect like terms on each side of equals sign. Simplify. Square both sides again.

More Solving Radical Equations Check: √ Try:

Radical equations with indexes greater than 2 can be solved using similar techniques. Check: After isolating the term containing the radical, raise each side of the equation to the power equal to the index of the radical. √

Domain and Range

Odd radical functions Domain and Range are ALL Real numbers

Practice Worksheet