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Creating New Information from a Functional Equation

Problem

Try it yourself before revealing the solution.

A function ff satisfies f(x)+2f(3−x)=x+6f(x)+2f(3-x)=x+6 for every real number xx.

Find f(1)f(1).

Need a hint?

The equation with x=1x=1 involves f(1)f(1) and f(2)f(2). What happens if you use x=2x=2?

Reveal solution
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What to remember

Use these observations when you meet a similar problem.

Why It Is Difficult

The first substitution does not determine f(1)f(1) by itself. The learner must notice that the transformation x↦3−xx\mapsto3-x exchanges the two relevant inputs and use the functional equation again.

Common Mistakes

Trying to guess a formula for ff, stopping after obtaining one equation in two unknowns, or choosing unrelated substitutions instead of following the input relationship already present.

Practice Problems

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Practice Problem 1

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Practice Problem 2

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