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Function Composition quiz

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  • What does function composition involve?
    Function composition involves replacing the variable in one function with another function, creating a new function.
  • How do you evaluate a function at a specific value?
    You substitute the variable in the function with the given value and simplify to get a number.
  • In the notation f(g(x)), which function is the inside function?
    In f(g(x)), g is the inside function.
  • What is the result of composing f(x) = x^2 + 3x - 10 and g(x) = x - 2 as f(g(x))?
    The result is (x - 2)^2 + 3(x - 2) - 10, which simplifies to x^2 - x - 12.
  • How do you find f(g(x)) if f(x) = x + 4 and g(x) = x^2 - 3?
    Replace x in f(x) with g(x), giving f(g(x)) = (x^2 - 3) + 4 = x^2 + 1.
  • What is the difference between f(g(x)) and g(f(x))?
    f(g(x)) means plugging g(x) into f(x), while g(f(x)) means plugging f(x) into g(x); the order matters.
  • What are two methods for evaluating a composed function at a number?
    You can either compose the functions first and then substitute the number, or evaluate the inside function at the number and then plug that result into the outside function.
  • If f(x) = x^2 and g(x) = x - 1, what is f(g(3))?
    First, g(3) = 2, then f(2) = 4, so f(g(3)) = 4.
  • Why might the shortcut method for evaluating composed functions not always work?
    The shortcut only works if you are not required to find the general form of f(g(x)) before evaluating at a specific value.
  • What is the first step in finding the domain of a composed function f(g(x))?
    First, find the domain restrictions of the inside function g(x).
  • How do you find additional domain restrictions when composing functions?
    After finding the domain of g(x), substitute g(x) into f(x) and find any new restrictions that arise from the composition.
  • 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, ∞).
  • What is function decomposition?
    Function decomposition is expressing a given function as the composition of two simpler functions.
  • How can you decompose h(x) = sqrt(x - 2) into two functions f and g?
    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? Explain.
    No, there are multiple valid ways to decompose a function, as long as composing the chosen functions gives the original function.