# SAT II Math I : Solving Other Functions

## Example Questions

### Example Question #1 : Solving Other Functions

Simplify:

You may assume that  is a nonnegative real number.

Explanation:

The best way to simplify a radical within a radical is to rewrite each root as a fractional exponent, then convert back.

First, rewrite the roots as exponents.

Multiply the exponents, per the power of a power rule:

### Example Question #2 : Solving Other Functions

Define functions  and .

for exactly one value of  on the interval .

Which of the following statements is correct about ?

Explanation:

Define

Then if ,

it follows that

,

or, equivalently,

.

By the Intermediate Value Theorem (IVT), if  is a continuous function, and  and  are of unlike sign, then  for some . As a polynomial,  is a continuous function, so the IVT applies here.

Evaluate  for each of the following values: :

Only in the case of  does it hold that  assumes a different sign at both endpoints - . By the IVT, , and , for some .