MHB Find the constants given the domain and range

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To find constants B and C for the function y = f(x) with a domain of 1 ≤ x ≤ 6 transformed to 8 ≤ x ≤ 9, B must be positive, allowing for algebraic manipulation to create two equations based on the endpoints. For the second part, to adjust the range of Af(x) + D from −3 ≤ y ≤ 5 to 0 ≤ y ≤ 1, constants A and D can be determined by setting up equations that align the new range with the original. The discussion emphasizes the need for algebraic relationships to solve for the constants effectively. Understanding the composition of functions is crucial for these transformations. The thread highlights the importance of correctly applying function transformations in pre-calculus.
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Suppose you have a function y = f(x) such that the domain of f(x) is 1 ≤ x ≤ 6 and the range of f(x) is −3 ≤ y ≤ 5.

a) Find constants B and C so that the domain of f(B(x − C)) is 8 ≤ x ≤ 9
B=
C=

b) Find constants A and D so that the range of Af(x) + D is 0 ≤ y ≤ 1
A=
D=

I'm working on composition of functions and completely lost at this point.
 
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Hello and welcome to MHB, bcast!

I have moved your topic from the Analysis forum as this is a Pre-calculus topic.

For the first problem, I would begin with the function's new domain:

$$8\le x\le9$$

Now, assuming $B$ is positive, can you algebraically get $B(x-C)$ in the middle, and then equating the end-points to the originals, you will have two equations in two unknowns?
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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