Exam-style question
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Let f(x) = x + 2 and g(y) = y^3, with both functions having domain and co-domain ℝ. Derive an expression for g ○ f and use it to calculate (g ○ f)(2). Show the intermediate result from applying f.
Model answer
What a good answer should say
- Apply f first: f(x) = x + 2.
- Applying g to this result gives (g ○ f)(x) = g(f(x)) = g(x + 2) = (x + 2)^3.
- For x = 2, f(2) = 4, and then g(4) = 4^3 = 64.
- Therefore, (g ○ f)(2) = 64.
Explanation
Why this works
The composition requires the output of f to become the input of g. Substituting f(x) into g gives the expression (x + 2)^3, and evaluating the intermediate value at x = 2 gives the final result 64.
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