the function for the perimeter of a rectangle is given as P=4x+6

a) Give an expression for the length of the rectangle in terms of the width.

b) If the width is half the length, state a function for both the width and length describing the domain for which these fuctions exist.

a) To find the expression for the length of the rectangle in terms of the width, we can start by rearranging the given formula for the perimeter.

P = 4x + 6

Since the perimeter of a rectangle is given by the formula P = 2(length + width), we can equate these two expressions and solve for the length (which in this case is represented by x):

2(length + width) = 4x + 6

Now, let's substitute the width (represented by w) into the equation:

2(length + w) = 4x + 6

Since the width is w, we have:

2(length + w) = 4x + 6

Divide both sides of the equation by 2 to isolate the length:

length + w = 2x + 3

Subtract the width (w) from both sides of the equation:

length = 2x + 3 - w

So, the expression for the length of the rectangle in terms of the width is:

length = 2x + 3 - w

b) Let's denote the width as w and the length as l. We are given that the width is half the length, so we can express this relationship as:

w = 0.5l

From the equation in part a), we had:

length = 2x + 3 - w

Now, substitute the value of w from the given relationship:

length = 2x + 3 - 0.5l

To define the domain for which these functions exist, we need to ensure that the values of length and width are positive since they represent physical measurements. Additionally, we need to consider that the width cannot be greater than or equal to the length since the width is half the length. Therefore, we have the following conditions:

l > 0
w > 0
w < l

For the functions to exist, we need positive lengths and widths, and the width must be smaller than the length.