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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 aa directly. The learner must identify two possible intermediate inputs of ff and then follow each possibility back through gg.

Common Mistakes

Keeping only one of x=4x=4 and x=−4x=-4, treating the auxiliary variable xx as though it were aa, or unnecessarily finding an explicit formula for ff.

Practice Problems

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Practice Problem 1

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Practice Problem 2

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Practice Problem 3

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Practice Problem 4

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Practice Problem 5

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Practice Problem 6

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