# Precalculus : Relations and Functions

## Example Questions

### Example Question #2 : Triangles

Let ABC be a right triangle with sides  = 3 inches,  = 4 inches, and  = 5 inches. In degrees, what is the  where  is the angle opposite of side ?

Explanation:

We are looking for . Remember the definition of  in a right triangle is the length of the opposite side divided by the length of the hypotenuse.

So therefore, without figuring out  we can find

### Example Question #1 : Simplifying Trigonometric Functions

Simplify the function below:

Explanation:

We need to use the following formulas:

a)

and

b)

We can simplify as follows:

Simplify

Explanation:

.  Thus:

Simplify

Explanation:

and

.

### Example Question #4 : Trigonometric Equations

If  , find the value of the function .

1

Explanation:

We can write:

We can also write:

Now we can substitute the values of (sinx - cosx) and (sinx * cosx) in the obtained function:

### Example Question #11 : Relations And Functions

Identifying a function.

Which of the following is a function.

Explanation:

The only relation listed that doesn't map more than one dependent variable value for some independent variable value is h(x).  Another way of saying this is that each value in the domain of h corresponds to a distinct value in its range.  h(x) is the only one-to-one relation, and so is the only option that is a function.

### Example Question #12 : Relations And Functions

Evaluating a function.

Evaluate , when

.

Explanation:

Plug in -3 for x in the function rule:

.

### Example Question #13 : Relations And Functions

Finding inverse functions.

Given,

find

.

Explanation:

To find an inversefunction find the input in terms of the output.  In other words, solve the function for the dependent variable.

First, set h(x) = y.  Now solve for y.

Multiply the denominator on both sides

Distribute y

Rearrange to get only x terms on one side.

Factor out x on the left side.

Divide by (y-5) on both sides to get x by itself.

The final step is to recognize that the independent variable is now the dependent variable.  To put this into proper notation switch x and y.  The inverse function is

.

### Example Question #14 : Relations And Functions

Finding an inverse function.

Given

find

.

.

.

Explanation:

Finding an inverse function is essentially finding the output in terms of the input.  To do this switch the independent and dependent variables.  First set

.

Then

Now, to switch the role of the variables solve for x in terms of y.

or

So, y is now the independent variable, and x is the dependent variable.  The final step is to write the inverse function in proper notation.

Set  , and .

This gives

.

### Example Question #15 : Relations And Functions

Which function below correctly translates  four units to the right and two units down?