Step 1: Find the minimum and maximum range for each equation..

Let's start with ..

I will plug in , I get . This is in my domain, which means it's the lowest value for the range. The maximum value is



Minimum range: 

Maximum range: 

Step 2: Write the solution in interval notation form:

 always get parentheses

 get brackets because they are in my range...

Answer:

Step 1: Rearrange the terms into the form y=mx+b. Move the  to the other side. 

Step 2: Move the 4 to the other side.



Step 3: When the line crosses the y-axis, the x value is zero. We will plug in  for x and find the y value.




So, the point where this line crosses the y-axis is 

Step 1: Look at the equation.. . The cube-root outside of the function determines what the answer is..

The function is a cube-root function.

Note:

Square function, 
Cube function, 
Rational function,  (if )

Step 1: To classify any function, find the degree of that function. The degree of a function is the highest exponent.

Step 2: Find degree of .

The degree is 3..

Step 3: Find what kind of function has degree of 3:

A cubic function has a degree of 3...

A square function has a degree of 2
A line has a degree of 1

Step 1: Define some basic graphs and their functions...

Line functions have an equation , where 
Parabola (Quadratic functions) have an equation  or  or , where 

Cubic functions have an equation , where 

Step 2: Determine what type of function is given...

The equation given is a parabola...

Quadratic functions have at least an  term.
Absolute Functions have .

Cube-Root functions have 

The function  is a cube-root function because it shares 

The minimum distance from the point to the line would be found by drawing a segment perpendicular to the line directly to the point. Our first step is to find the equation of the new line that connects the point to the line given in the problem. Because we know this new line is perpendicular to the line we're finding the distance to, we know its slope will be the negative inverse of the line its perpendicular to. So if the line we're finding the distance to is:

Then its slope is -1/3, so the slope of a line perpendicular to it would be 3. Now that we know the slope of the line that will give the shortest distance from the point to the given line, we can plug the coordinates of our point into the formula for a line to get the full equation of the new line:

Now that we know the equation of our perpendicular line, our next step is to find the point where it intersects the line given in the problem:

So if the lines intersect at x=0, we plug that value into either equation to find the y coordinate of the point where the lines intersect, which is the point on the line closest to the point given in the problem and therefore tells us the location of the minimum distance from the point to the line:

So we now know we want to find the distance between the following two points:

  and 

Using the following formula for the distance between two points, which we can see is just an application of the Pythagorean Theorem, we can plug in the values of our two points and calculate the shortest distance between the point and line given in the problem:

Which we can then simplify by factoring the radical:

The shortest distance from a point to a line is always going to be along a path perpendicular to that line. To be perpendicular to our line, we need a slope of .

To find the equation of our line, we can simply use point-slope form, using the origin, giving us

  which simplifies to .

Now we want to know where this line intersects with our given line. We simply set them equal to each other, giving us .

If we multiply each side by , we get .

We can then add  to each side, giving us .

Finally we divide by , giving us .

This is the x-coordinate of their intersection. To find the y-coordinate, we plug  into , giving us .

Therefore, our point of intersection must be .

We then use the distance formula  using  and the origin.

This give us .

Draw a line that connects the point and intersects the line at a perpendicular angle.  

The vertical distance from the point  to the line  will be the difference of the 2 y-values.  

The distance can never be negative.

Distance cannot be a negative number.  The function  is a vertical line. Subtract the value of the line to the x-value of the given point to find the distance.