A=l * wWe are given the length (x-3) and the width 8. Substituting this into the above formula, we can determine an expression for the area of the rectangle.
By replacing A for A(x) we can write this expression in function notation.
A(x)=8x-24
b The domain shows the possible values of x. Since the rectangles length is x-3 and the length has to be a positive value, we know that x-3 has to be greater than 0.
x-3>0
By solving this inequality we can determine the domain.
The domain is x>3. Notice that x cannot be 3, as this would give a length of 0 and consequently, no rectangle. In other words, given the domain of x>3, the range has to be greater than 0. Therefore, the domain and range can be written as follows.
Domain: &x>3
Range: &A(x)>0
Let's graph the function.
c To algebraically determine the inverse function of A(x) we write A(x) as y, interchange x and y, then isolate y.
y=8 x-24 ⇒ x=8 y-24By solving for y we can find the inverse function.
We can now replace y for A^(- 1)(x) to show the inverse of A(x). In the original function, x represented a value that determined the rectangle's length and A(x) represented the rectangle's area.
A^(- 1)(x)=1/8x+3
When inverting a function, items represented by one variable in the given function will be represented by the other variable in the inverse function. Therefore, x in the inverse function represents the rectangle's area and C^(- 1)(x) represents the value that determines the rectangle's length.
d
Since A^(- 1)(x) represents the value that determines the length, we know that A^(- 1)(x)>3. Similarly, since x represents the rectangle's area, we know that x>0. With this information we can graph the function.
The domain of A^(- 1)(x) represents the area of the rectangle which must always be positive. The range of A^(- 1)(x) represents the possible values of x in the expression x-3. The range is the set of all real numbers which are greater than 3.
e Remember that when inverting a function, the items represented by one variable in the given function will be represented by the other variable in the inverse function. The range of A(x) is therefore the domain of A^(- 1)(x), and the domain of A(x) is the range of A^(- 1)(x).
