1. The function f(x) is not defined for x = 0. It has the property

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The discussion focuses on the function f(x) defined by the equation f(x) + 2f(1/x) = 3x for all nonzero real numbers x. It also explores the specific form f(x) = (ax+b)/(cx+d), where a, b, c, and d are nonzero real numbers, and the properties f(19) = 19 and f(97) = 97. Participants emphasize isolating f(x) by substituting y = 1/x and identifying the unique number not in the range of f, which is determined to be -d/c.

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  • Familiarity with properties of inverse functions
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  • Learn about rational function transformations and their implications
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Mathematicians, students studying functional equations, and anyone interested in advanced algebraic concepts will benefit from this discussion.

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1. The function f(x) is not defined for x = 0. It has the property that for all nonzero real numbers x, f(x) + 2f(1/x) = 3x. Find all values of a such that f(a) = f(-a)

2. The function f is defined by f(x) = (ax+b)/(cx+d), where a, b, c, and d are nonzero real numbers, and has the properties: f(19) = 19, f(97) = 97, and f(f(x) = x for all values of x except -d/c. Find the unique number that is not in the range of f.

First time seeing this. Somehow tell me how to approach it, I don't need the answer.
 
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1. Try to isolate f(x) by using the equation you have by setting ing y = 1/x.

2. Obviously, since f(f(x)) = x for all x /= -d/c, the number have to be -d/c. How can you find this value?
 

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