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Cogito

AP Precalculus · Chapter 4 · Lesson 3

Practice Worksheet

Composition and Decomposition

Building functions, and taking them apart.

12 problems · about 22 minutes · AP Precalculus 1.13

NameDate

Figure — use these to answer the problems

The square root function has a restricted domain. Composing with it imposes that restriction on the inside.
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y = 1√x + 0
  • Point(4, 2)
a, coefficient1
Point on the curve4
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. f(x) = 3x and g(x) = x + 2. What is f(g(4))?

    Answer

  2. Why find a composite domain before simplifying?

    • Simplifying can hide a restriction the original composite had.
    • Because simplifying is difficult.

  3. f(x) = x + 1, g(x) = 3x. What is f(g(2))?

    Answer

  4. Same functions. What is g(f(2))?

    Answer

  5. f(x) = √x, g(x) = x − 9. Smallest x in the domain of f(g(x))?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. f(x) = 1 ÷ x, g(x) = x − 2. Which x must be excluded from f(g(x))?

    Answer

  2. For √(3x + 1), what is the inner function at x = 5?

    Answer

  3. f(x) = x², g(x) = x + 1. What is f(g(3))?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Put the steps for finding a composite domain in order.

    Write 1 to 4 in the boxes to put these in order.

    • Find the domain of the inner function.
    • Require the inner output to lie in the outer domain.
    • Combine both restrictions.
    • Only then simplify the expression.

  2. f(x) = 2x, g(x) = x². What is f(g(3))?

    Answer

  3. The Composition

    f(x) = 2x and g(x) = x + 3. What is f(g(4))?

    Answer

  4. The Domain

    f(x) = √x and g(x) = x − 5. What is the smallest x in the domain of f(g(x))?

    Answer