Notes 9-5 and 8-8

Objective:

I will be able to:

Solve radical equation by raising both sides to the reciprocal power.

Solve a radical inequality.

Solve a radical equation for a specific variable.

Algebraically find the inverse of a cubic, or radical function (i.e. square root, cube root) and state it in inverse notation.

Determine the domain and range of the inverse of a cubic, or radical function (i.e. square root, cube root).

Apply the horizontal line test (HLT) to determine if the inverse of a relation is a function.

Interpret a graph to determine the following:

  • Is the relation a function?
  • Is the inverse a function?

SOLVE RADICAL EQUATION BY RAISING BOTH SIDES TO THE RECIPROCAL POWER.

Ex 1:Ex 2:

Remember to check for EXTRANEOUS ROOTS,

by substituting back into the ORGINAL EQUATION.

X=-4is extraneous b/c

X=5

SOLVE A RADICAL INEQUALITY.

Ex 3: Refer to page 630 Example 5

Note the following:

a. In the first solution, the calculator was used. Y1 was the left side of the inequality and Y2 was the right side. Since you want , look in the table for the values where that is true. You will see that whenever , then Y1 Y2. It may be necessary to use the “Calculate Intersection” feature to determine the exact value where the two functions are equal.

b. In the second solution, each side of the inequality was raised to the second power, and it was solved just like an equation. However, be careful if you multiply or divide by a negative!

This method requires a second step to ensure that the radicand is defined. Therefore, you must set . By doing so, you get that must ALSO be true.

The final answer is that .

c. You should be able to solve radical inequalities by BOTH methods.

SOLVE A RADICAL EQUATION FOR A SPECIFIC VARIABLE.

Ex. 4: Solve for v—Isolate v.

ALGEBRAICALLY FIND THE INVERSE OF A CUBIC, OR RADICAL FUNCTION

This is the 4 step process of:

a. replace f(x) with y

b. swap x and y

c. solve for y

d. replace y with f-1(x) REMEMBER f-1(x) is inverse notation and stands for the inverse of f(x).

Ex. 5: f(x)=27x3+10

y = 27x3+10

x = 27y3+10

APPLY THE HORIZONTAL LINE TEST (HLT) TO DETERMINE IF THE INVERSE OF A RELATION IS A FUNCTION.

We apply the VERTICAL line test to a relation to determine if it is a function. In much the same way, we can apply the HORIZONTAL line test to a relation, to determine if the inverse is a function.

Ex. 6: Refer to page 690 –example 1A. As you pass a HORIZONTAL line from , we see that the HORIZONTAL line contains only 1 pt of the relation at any given time. Therefore, this relationPASSES the HORIZONTAL line test and we conclude that the INVERSE will be a FUNCTION.

Refer to page 690—example 1B. As you pass a HORIZONTAL line from , we see that the HORIZONTAL line contains more than 1 pt of the relation at various times. Therefore, this relation FAILS the HORIZONTAL line test and we conclude that the INVERSE is NOT a FUNCTION.