Question

Bronze is an alloy or mixture of the metals copper and tin. The properties of bronze depends on the percentage of copper in the mix. A chemist decides to study the properties of a given alloy of bronze as the proportion of copper is varied. She starts with 7 kg of bronze that contain 3.5 kg of copper and 3.5 kg of tin and either adds or removes copper. Let f(x) be the percentage of copper in the mix if x kg of copper are add (x>0) or removed (x<0).
(
a. State the domain and range of f (assume that the chemist wants a non-negative amount of both copper and tin).
Domain: Range: What do each of the answers mean in the context of bronze?
The domain tells us The range tells us (
b. Write a formula in terms of x for f(x). (
c. If the formula you found in part (
b. was not intended to represent the percentage of copper in an alloy of bronze, but instead simply defined an abstract mathematical function, what would be the domain and range of this function?
Domain: Range:

Answer 1

(a)

Because there is 3.5 kg of copper in the original mix, then the maximum amount of copper that can be removed is 3.5 kg, in that case, x would be equal to -3.5 and the percentage of copper would be 0%. Then, (-3.5, infinity) interval is the domain. When 3.5 kg of copper are removed, f(x) = 0%, and in the limit when x tends to infinity, f(x) = 100%. So, the range is 0% to 100%.      

(b)

The percentage of copper in the mix, f(x) = 100*(3.5 + x)/(7 + x)

where x is kg of copper that are add (x>0) or removed (x<0)

(c)

In this case, the domain of the function is every real number except -7, i. e., (-infinity, -7) U (-7, infinity) interval.  

The range of a rational function where the grade of the polynomial in the numerator is less or equal than the grade of the polynomial in the denominator is all real numbers, i. e., (-infinity, infinity) interval.

Answer 2

A. The domain is any real number.
The range is f(x) ≥ 0.5 (or 50%)

b. The formula is
f(x) = (3.5 + x) / (7 + x)

c. The domain is any number except -7. The range is any real number greater than 0.