Since the piecewise-defined function  is defined two different ways, one for negative numbers and one for nonnegative numbers, examine both definitions and determine each partial range separately;  the union of the partial ranges will be the overall range.

If , then 

Since 

,

applying the properties of inequality,

Therefore, on the portion of the domain comprising nonpositive numbers, the partial range of  is the set .

If , then 

Since 

,

applying the properties of inequality,

Therefore, on the portion of the domain comprising positive numbers, the partial range of  is the set .

The overall range is the union of these partial ranges, which is .

First, we must determine whether  exists.

A quadratic function has a parabola as its graph; this graph decreases, then increases (or vice versa), with a vertex at which the change takes place. 

 exists if and only if, if , then - or, equivalently, if there does not exist  and  such that , but . This will happen on any interval on which the graph of  constantly increases or constantly decreases, but if the graph changes direction on an interval, there will be  such that  on this interval. The key is therefore to determine whether the interval to which the domain is restricted contains the vertex.

The -coordinate of the vertex of the parabola of the function

is .

The -coordinate of the vertex of the parabola of  can be found by setting :

.

The vertex of the graph of  without its domain restriction is at the point with -coordinate 4. Since , the vertex is in the interior of the domain; as a consequence,  does not exist on .

A function of the form  is a linear function and is either constantly increasing or constantly decreasing. Therefore, we can simply note that if

,

as stated in the domain, then, multiplying both sides by , remembering to switch the symbol since we are multiplying by a negative number:

Add 12 to both sides:

Replacing, we see that 

,

so the range of  is .

If , it follows by applying the properties of inequality that:

Multiply both sides by , which must be positive by closure:

, 

that is, 

Also,by closure, , so

This makes the correct range .

Substitute 6x for x in the f(x) function and simplify.

Since we are given the functions x-intercepts, we can write the equation as , now we need to use FOIL.

Start with .

Now we do

What this question is asking for is to approximate the slope of the line of best fit. To calculate the slope, we need to find find two points on the line. The best points to pick out is , and . Now we can find the rate of change by using the following equation.

, where  are points on the line.

Break up the problem into parts.

Two times a number:  

 The sum of two times a number and forty:  

Is equal to sixteen:  

Combine the terms to form the equation.

The answer is:  

Split up the question into parts.

The square of a number:  

Three less than the square of a number:  

Is eleven:  

Combine the terms.

The answer is:  

Write the following sentence by parts.

A number squared:  

The difference of six and a number squared:  

Is four:  

Combine the parts to write an equation.

The answer is: