# Advanced Geometry : How to graph a function

## Example Questions

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### Example Question #1 : Calculating The Length Of A Chord

The chord of a  central angle of a circle with area  has what length?

Explanation:

The radius  of a circle with area  can be found as follows:

The circle, the central angle, and the chord are shown below:

By way of the Isosceles Triangle Theorem,  can be proved equilateral, so , the correct response.

### Example Question #2 : Calculating The Length Of A Chord

The chord of a  central angle of a circle with area  has what length?

Explanation:

The radius  of a circle with area  can be found as follows:

The circle, the central angle, and the chord are shown below, along with , which bisects isosceles

We concentrate on , a 30-60-90 triangle. By the 30-60-90 Theorem,

and

The chord  has length twice this, or

### Example Question #8 : Calculating The Length Of A Chord

The chord of a  central angle of a circle with circumference  has what length?

Explanation:

A circle with circumference  has as its radius

.

The circle, the central angle, and the chord are shown below:

By way of the Isosceles Triangle Theorem,  can be proved equilateral, so , the correct response.

### Example Question #1 : How To Graph A Function

What is the domain of ?

all real numbers

all real numbers

Explanation:

The domain of the function specifies the values that  can take.  Here, is defined for every value of , so the domain is all real numbers.

### Example Question #2 : How To Graph A Function

What is the domain of ?

Explanation:

To find the domain, we need to decide which values  can take.  The  is under a square root sign, so  cannot be negative.   can, however, be 0, because we can take the square root of zero.  Therefore the domain is .

### Example Question #2 : How To Graph A Function

What is the domain of the function ?

Explanation:

To find the domain, we must find the interval on which is defined.  We know that the expression under the radical must be positive or 0, so is defined when .  This occurs when and .  In interval notation, the domain is .

### Example Question #4 : How To Graph A Function

Define the functions  and  as follows:

What is the domain of the function  ?

Explanation:

The domain of  is the intersection of the domains of  and and  are each restricted to all values of  that allow the radicand  to be nonnegative - that is,

, or

Since the domains of  and  are the same, the domain of  is also the same. In interval form the domain of  is

### Example Question #2 : How To Graph A Function

Define

What is the natural domain of ?

Explanation:

The radical in and of itself does not restrict the domain, since every real number has a real cube root. However, since the expression  is in a denominator, it cannot be equal to zero, so the domain excludes the value(s) for which

27 is the only number excluded from the domain.

### Example Question #6 : How To Graph A Function

Define

What is the natural domain of  ?

Explanation:

Since the expression  is in a denominator, it cannot be equal to zero, so the domain excludes the value(s) for which . We solve for  by factoring the polynomial, which we can do as follows:

Replacing the question marks with integers whose product is  and whose sum is 3:

Therefore, the domain excludes these two values of .

### Example Question #7 : How To Graph A Function

Define .

What is the natural domain of ?

Explanation:

The only restriction on the domain of  is that the denominator cannot be 0. We set the denominator to 0 and solve for  to find the excluded values:

The domain is the set of all real numbers except those two - that is,

.

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