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Following a Chain of Function Relationships

Problem

Try it yourself before revealing the solution.

Functions ff and gg satisfy f(2x+1)=x2+3f(2x+1)=x^2+3 and g(x)=2x−3g(x)=2x-3 for all real xx.

Given that f(g(a))=19f(g(a))=19, find the sum of all possible values of aa.

Need a hint?

First determine which inputs of ff can produce the output 19. Then match those inputs with g(a)g(a).

Reveal solution
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What to remember

Use these observations when you meet a similar problem.

Why It Is Difficult

The final output does not determine a directly. The learner must identify every possible input of f and then use the relationship with g.

Common Mistakes

Keeping only one solution, confusing the auxiliary variable with a, or unnecessarily finding an explicit formula for f.

Practice Problems

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Practice Problem 1

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Practice Problem 2

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