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Function Composition quiz

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  • What does it mean to compose two functions, such as f(g(x))?
    Composing two functions means replacing the variable in one function with another function, resulting in a new function.
  • If f(x) = x^2 + 3x - 10 and g(x) = x - 2, what is f(g(x))?
    f(g(x)) = (x - 2)^2 + 3(x - 2) - 10, which simplifies to x^2 - x - 12.
  • How do you evaluate a composed function at a specific value using the two-step method?
    First, evaluate the inner function at the given value, then substitute this result into the outer function.
  • What is the shortcut method for evaluating f(g(a))?
    Evaluate g(a) first, then plug that result into f(x) to get f(g(a)).
  • Why might you not always use the shortcut method for evaluating composed functions?
    Because sometimes you are required to find the general form of f(g(x)) before substituting a value.
  • If f(x) = x + 4 and g(x) = x^2 - 3, what is f(g(x))?
    f(g(x)) = (x^2 - 3) + 4 = x^2 + 1.
  • If f(x) = x + 4 and g(x) = x^2 - 3, what is g(f(x))?
    g(f(x)) = (x + 4)^2 - 3, which expands to x^2 + 8x + 16 - 3 = x^2 + 8x + 13.
  • How do you find the domain of a composed function f(g(x))?
    Find the domain restrictions for both g(x) and f(g(x)), then combine these restrictions for the overall domain.
  • If f(x) = 1/(x-2) and g(x) = sqrt(x), what is the domain of f(g(x))?
    The domain is all x ≥ 0 except x ≠ 4, so [0, 4) ∪ (4, ∞).
  • Why must you consider the domain of both the inner and outer functions when composing functions?
    Because the input must be valid for the inner function and the result must be valid for the outer function.
  • What is function decomposition?
    Function decomposition is expressing a single function as the composition of two functions, f and g.
  • Given h(x) = sqrt(x - 2), how can you decompose h(x) into f(g(x))?
    Let g(x) = x - 2 and f(x) = sqrt(x), so h(x) = f(g(x)).
  • Is there only one correct way to decompose a function?
    No, there are multiple correct ways to decompose a function, depending on how you define f(x) and g(x).
  • What is a common strategy for decomposing a function involving a square root?
    Set g(x) equal to the expression inside the square root and f(x) equal to the square root function.
  • How can creativity play a role in function decomposition?
    You can choose different expressions for g(x) and adjust f(x) accordingly, as long as their composition gives the original function.