# SAT II Math II : Solving Functions

## Example Questions

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### Example Question #1 : Solving Functions

If , what must  be?

Explanation:

Replace the value of negative two with the x-variable.

There is no need to use the FOIL method to expand the binomial.

### Example Question #2 : Solving Functions

Let .  What is the value of ?

Explanation:

Substitute the fraction as .

Multiply the whole number with the numerator.

Convert the expression so that both terms have similar denominators.

### Example Question #3 : Solving Functions

If , what must  be?

Explanation:

A function of x equals five.  This can be translated to:

This means that every point on the x-axis has a y value of five.

Therefore, .

### Example Question #4 : Solving Functions

Rewrite as a single logarithmic expression:

Explanation:

Using the properties of logarithms

and ,

simplify as follows:

### Example Question #5 : Solving Functions

Simplify by rationalizing the denominator:

Explanation:

Multiply the numerator and the denominator by the conjugate of the denominator, which is . Then take advantage of the distributive properties and the difference of squares pattern:

### Example Question #6 : Solving 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.

Then convert back to a radical and rationalizing the denominator:

### Example Question #7 : Solving Functions

Let .   What is the value of ?

Explanation:

Replace the integer as .

Evaluate each negative exponent.

Sum the fractions.

### Example Question #8 : Solving Functions

Find :

Explanation:

Square both sides to eliminate the radical.

Divide by negative three on both sides.

### Example Question #9 : Solving Functions

If , what is the value of ?

Explanation:

Substitute the value of negative three as .

The terms will be imaginary.  We can factor out an  out of the right side.  Replace them with .

### Example Question #10 : Solving Functions

A baseball is thrown straight up with an initial speed of 60 feet per second by a man standing on the roof of a 100-foot high building. The height of the baseball in feet as a function of time   in seconds is modeled by the function

To the nearest tenth of a second, how long does it take for the baseball to hit the ground?

Explanation:

When the baseball hits the ground, the height is 0, so we set . and solve for .

This can be done using the quadratic formula:

Set :

One possible solution:

We throw this out, since time must be positive.

The other:

This solution, we keep. The baseball hits the ground in 5 seconds.

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