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RADICAL FUNCTIONS
INTRO TO RADICAL EXPRESSIONS Review Square Roots Extend to Cubed Roots Rationalizing the Denominator of an Expression Working with Rational Exponents ADD/SUBTRACT/MULTIPLY/DIVIDE Simplifying Radical Expressions within Operations
SOLVING RADICAL EQUATIONS Using Inverse Operations Extraneous Solutions (Check Answers) GRAPHING RADICAL FUNCTIONS Domain Transformations
APPLICATIONS
INTRODUCTION TO RADICAL EXPRESSIONS *Review โ Simplify each of the following Radical Expressions. Assume all variables are positive.
1. โ50 2. โ32๐ฅ5 3. โ27๐3๐43
* Fill a value into each radical so that the result is a whole number solution, then write the solution. A couple of problems are already filled in as examples.
4. โ1212
= 11 5. โ____ = ________ 6. โ_____2 = ________
7. โ643
= 4 because 4 โ 4 โ 4 = 64 8. โ_____3 = ________ 9 โ_____4 = ________
* Review โ Rationalize the denominator in each of the following. Again, the first is provided as an example.
10. ๐
โ๐๐
๐
โ๐๐ โ
โ๐๐
โ๐๐ =
๐โ๐๐
โ๐๐ =
๐โ๐๐
๐
11. ๐
โ๐๐ 12.
๐
โ๐
* Rationalize the denominator in each.
13. ๐
โ๐๐
๐
โ๐๐ โ
โ๐๐
โ๐๐ =
๐โ๐๐
โ๐๐ =
๐โ๐๐
๐= ๐โ๐
๐
14. Written Response: Why did the example at the left multiply by one
in the form of โ๐๐
โ๐๐ ?
15. ๐
โ๐๐ 16.
๐๐
โ๐๐
OPERATIONS WITH RADICAL EXPRESSIONS * Intro/Review โ Working with Rational Exponents A Rational Exponent is a Root and a Power Combined.
Example: (8)23 means (โ8
3)
2. Do you see where the 2 and the 3 from the rational exponent are in radical expression?
So (8)23 = (โ8
3)
2= (2)2 = 4
*Review โ Simplify each of the following Radical Expressions.
1. (27)23 2. (16)
32 3. (125)
13
*Review โ Write each in radical form and then simplify. Assume all variables are positive.
4. (27๐3๐4)1
2 5. (๐๐๐๐๐๐)๐
๐ 6. (๐๐๐๐๐๐)๐
๐
*Simplify each based on the mathematical operation. Assume all variables are positive.
7. (๐ + ๐โ๐) โ (๐ + โ๐๐) 8. ๐โ๐(๐โ๐ + ๐โ๐)
9. (๐ โ ๐โ๐)(๐ + ๐โ๐)
10. โ๐๐๐๐๐๐โ (๐๐๐๐๐)
๐
๐ โ โ๐๐๐
11. โ๐๐๐๐๐๐
(๐๐๐๐๐)๐๐
12. (๐ + โ๐)(โ๐ + โ๐)
*Review โ *Simplify each based on the mathematical operation. Assume all variables are positive.
1. โ๐๐๐ โ โ๐๐๐ 2. (๐โ๐ โ ๐) โ (๐ โ โ๐) 3. โ๐
โ๐
4. โ๐๐๐๐๐๐๐๐ โ (๐๐๐๐๐๐)๐
๐ 5. (๐โ๐ โ ๐)(๐ โ โ๐) 6.
โ๐๐๐
โ๐๐๐๐
SOLVING RADICAL EQUATIONS Notes:
To solve a radical equation we must get rid of the radicals wrapped around the variable.
To undo a radical, square both sides of the equation after the radical is isolated.
The Steps: 1. Isolate a radical term, 2. Square both sides, 3. Continue until all radicals are gone, 4. Solve equation, 5. Check answers in original equation.
*Solve each radical equation.
1. โ๐ฅ โ 3 = 2 2. โ๐ฅ + 2 = 4โ๐ฅ + 1 3. โ๐ฅ2 โ 16 โ 3 = 0
MIXED EXAMPLES WITH RADICAL EXPRESSIONS/EQUATIONS *Simplify each Radical Expression. Also, rationalize each denominator in necessary. Solve each equation for x.
1. โ5
8 2.
3
5โโ7 3. โ๐ฅ โ 3 โ โ๐ฅ = 3
SOLVING RADICAL EQUATIONS * Solve each equation for the indicated unknown. Check each by graphing.
1. โ๐ฅ โ 3 + 6 = 5 2. โ9๐ฅ2 + 4 = 3๐ฅ + 2 3. โ4 โ 2๐ก โ ๐ก2 = ๐ก + 2
4. โ๐ฅ + 13
โ 3 = 4
MIXED EXAMPLES WITH RADICAL EXPRESSIONS/EQUATIONS *Simplify each Radical Expression. Also, rationalize each denominator in necessary. Solve each equation for x.
1. 3+5โ2
4โ3โ10 2. โ2๐ฅ + 5 = ๐ฅ โ 5
* Functions Name ______________________________ Period _______
Evaluating Functions Operations with Functions Composite Functions Difference Quotient
Transformations of Radical Functions Inverse Functions Composites of Inverses
Given: ๐(๐) = ๐๐ + ๐ ๐(๐) = ๐ + ๐ ๐(๐) = โ๐ โ ๐ ๐(๐) = ๐๐ + ๐๐ ๐(๐) = ๐๐ โ ๐
Evaluate each.
1. ๐(3)
2. ๐(โ2) 3. โ(22)
4. ๐(โ10)
5. ๐(5) 6. ๐(โ5)
Perform each operation and simplify the expression. State any domain restrictions when necessary.
7. (โ + ๐)(๐ฅ)
8. (๐ โ ๐)(๐ฅ) 9. (๐ โ ๐)(๐ฅ)
10. (๐ โ โ)(๐ฅ)
11. (โ
๐) (๐ฅ) 12. (
๐
๐) (๐ฅ)
Evaluate and Simplify each Composite Function.
13. ๐(๐(3))
14. โ(๐(5)) 15. ๐(๐(โ1))
16. (โ ๐ ๐)(โ2)
17. (๐ ๐ ๐)(3) 18. ๐(๐(๐ฅ))
19. ๐(๐(๐ฅ))
20. ๐(๐(๐ฅ)) 21. โ(๐(๐ฅ))
Extra Practice
Given: ๐(๐) = ๐๐ ๐(๐) = ๐ + ๐ ๐(๐) = โ๐๐ + ๐ ๐(๐) = ๐๐ โ ๐๐ ๐(๐) = ๐๐ โ ๐
Evaluate each.
1. ๐(3)
2. ๐(โ2) 3. โ(22)
4. ๐(โ10)
5. ๐(5) 6. ๐(โ5)
Perform each operation and simplify the expression. State any domain restrictions when necessary.
7. (โ + ๐)(๐ฅ)
8. (๐ โ ๐)(๐ฅ) 9. (๐ โ ๐)(๐ฅ)
10. (๐ โ โ)(๐ฅ)
11. (โ
๐) (๐ฅ) 12. (
๐
๐) (๐ฅ)
Evaluate and Simplify each Composite Function.
13. ๐(๐(3))
14. โ(๐(5)) 15. ๐(๐(โ1))
16. (โ ๐ ๐)(โ2)
17. (๐ ๐ ๐)(3) 18. ๐(๐(๐ฅ))
19. ๐(๐(๐ฅ))
20. ๐(๐(๐ฅ)) 21. โ(๐(๐ฅ))
Evaluating Piecewise Functions Given:
๐(๐) = {๐ + ๐ ๐๐ โ๐ โค ๐ < ๐
๐๐ โ ๐ ๐๐ ๐ โค ๐ < ๐๐ ๐(๐) = {
๐ ๐๐ โ๐ โค ๐ < 1๐๐ ๐๐ ๐ โค ๐ โค ๐
โ๐
๐๐ + ๐ ๐๐ ๐ > 3
๐(๐) = {
๐ โ ๐ ๐๐ โ๐ < ๐ < โ๐๐ ๐๐ โ๐ โค ๐ < ๐
๐๐ + ๐โ๐
๐๐๐๐
๐ โค ๐ โค ๐๐ > ๐
Evaluate each function.
1. ๐(๐)
2. ๐(โ๐) 3. ๐(๐) 4. ๐(โ๐)
5. ๐(๐)
6. ๐(โ๐) 7. ๐(๐) 8. ๐(๐)
9. ๐(๐)
10. ๐(๐(๐)) 11. ๐(๐(๐)) 12. ๐ (๐(๐(๐)))
Inverses
Notes โ Graphing Inverses The inverse of a function is a function that undoes the action of another function.
Graphically, and inverse is the reflection each point of the function across the line ๐ฆ = ๐ฅ.
The line ๐ฆ = ๐ฅ is the set of points where every x and y value are the same, i.e., the diagonal line from lower left to upper
right.
The reflection of a point is the point that is the same distance behind the reflected surface.
Ex. 1: Reflect each point across the line ๐ฆ = ๐ฅ.
Label the reflection of point A as Aโ, etc.
A(4, 3)
B(7, 4)
C(-5, -2)
D(-3, 6)
E(4, 4)
Ex. 2: Graph the inverse of ๐ฆ = 3๐ฅ + 1 by filling in the table for the equation, plotting the pairs of points, drawing a line
connecting these points, and then reflecting each point. Conclude by connecting the reflected points with a line.
x y
-2
-1
0
1
2
Practice
1. Graph the inverse of ๐ฆ = 2๐ฅ โ 4 by filling in the table for the equation, plotting the pairs of points, drawing a line
connecting these points, and then reflecting each point. Conclude by connecting the reflected points with a line.
x y
-2
-1
0
1
2
Given: ๐ฆ =1
2๐ฅ โ 3 ,
a. Fill-in the table below for the equation, b. plot the pairs of points, c. draw a line connecting these points, c. reflect
each point across the line ๐ฆ = ๐ฅ, d. Write the equation for the line created by the reflected points.
x y
-4
-2
0
2
4
Fill in the table below by using the points
from the reflected line.
x y
What do you notice when comparing this
table with the table above?
Algebraic Method of Finding the Inverse Equation
Composite of Inverse Functions
Introduction to the Difference Quotient Donโt be afraid, but this is the basic concept of the Derivative which is used in Calculus to look at Rates of Change.
Warm-Up Evaluate and Simplify each when ๐(๐ฅ) = ๐ฅ2 โ 5๐ฅ.
1. ๐(3)
2. ๐(7) 3. ๐(โ4) 4. ๐(1)
5. ๐(๐ฅ)
6. ๐(๐) 7. ๐(๐(3)) 8. ๐(๐(โ2))
9. ๐(๐ + โ)
10. ๐(๐โ1(๐ฅ)) 11. ๐(๐ + 5) 12. ๐(๐ฅ2)
You should recognize this:
๐ =๐๐ โ ๐๐
๐๐ โ ๐๐
Similar to this what is titled the Difference Quotient
๐ =๐(๐ + ๐) โ ๐(๐)
(๐ + ๐) โ ๐=
๐(๐ + ๐) โ ๐(๐)
๐
13. Evaluate and Simplify the Difference Quotient
for ๐(๐ฅ) = ๐ฅ2 โ 5๐ฅ.
14. Evaluate and Simplify the Difference Quotient
for ๐(๐ฅ) = 2๐ฅ2 + 3.
More on Domain Restrictions Notes:
Definition: Domain - the set of all possible input values (usually x), which allows the function formula to work.
(the x-values that are allowed to be used in a function)
Sometimes, rather than figuring out the x-values that are allowed, it is easier to figure out the x-values that
are not allowed and then state that you can use all x-values except for those.
Domain restrictions can occur within even roots, like square roots, and in the denominator of fractions. - We cannot take the square roots of negative numbers, therefore expressions within the square root must
be greater than or equal to 0.
- We cannot divide by zero, therefore expressions in the denominator cannot be equal to zero.
*State the domain for each of the following functions.
Ex. 1: ๐(๐ฅ) = โ๐ฅ + 1 Ex. 2: ๐(๐ฅ) =๐ฅ+4
๐ฅโ1 Ex. 3: ๐(๐ฅ) = ๐ฅ2 โ 3๐ฅ + 4
Practice
* State the domain for each of the following functions.
1. ๐(๐ฅ) = โ5๐ฅ โ 13 2. ๐(๐ฅ) =๐ฅ+1
(3๐ฅโ4)(๐ฅ+2) 3. ๐(๐ฅ) = 6๐ฅ3 + 6๐ฅ โ 1
4. ๐(๐ฅ) = โ5๐ฅ + 1 5. ๐(๐ฅ) =๐ฅโ9
(๐ฅโ1)(2๐ฅ+7) 6. ๐(๐ฅ) = ๐ฅ2 โ 3๐ฅ + 4
Inverses of Quadratic Functions
Algebraically determine the inverse of each quadratic function. Check for correctness using composites.
1. ๐(๐ฅ) = ๐ฅ2
2. ๐(๐ฅ) = (๐ฅ โ 7)2 3. โ(๐ฅ) = ๐ฅ2 + 5
Notes:
Keep in mind that a radical function is the inverse of a quadratic function.
You already should somewhat have a sense of this because you have learned that squaring and square rooting are inverse
operationsโฆthey undo one another.
Therefore when you see the graph of a radical function, it will look like half of a parabola on its side.
*State the domain of each radical function and then graph.
Ex. 1: ๐(๐ฅ) = โ๐ฅ Domain: ____________ Range: ____________
x
y
This is the graph of the parent function for radical functions. You will use your understanding of transformations to accurately graph the rest.
1. Graph ๐ฆ = โ๐ฅ โ 3 + 4 Domain: ____________ Range: ____________
2. Graph ๐ฆ = โโ๐ฅ + 2 Domain: ____________ Range: ____________
3. Graph ๐ฆ = 3โ๐ฅ Domain: ____________ Range: ____________
4. Graph ๐ฆ = 2โ๐ฅ + 5 โ 6 Domain: ____________ Range: ____________
5. Graph ๐ฆ = โโ๐ฅ + 3 Domain: ____________ Range: ____________