Mathematics » High School: Functions » Building Functions

Build a function that models a relationship between two quantities.

• 1. Write a function that describes a relationship between two quantities.^★

  Suggested tag: cc-9-12m-f-bf-1

  • Determine an explicit expression, a recursive process, or steps for calculation from a context.

    Suggested tag: cc-9-12m-f-bf-1a

  • Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

    Suggested tag: cc-9-12m-f-bf-1b

  • (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

    Suggested tag: cc-9-12m-f-bf-1c

• 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.^★

  Suggested tag: cc-9-12m-f-bf-2

Build new functions from existing functions.

• 3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

  Suggested tag: cc-9-12m-f-bf-3

• 4. Find inverse functions.

  Suggested tag: cc-9-12m-f-bf-4

  • Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x³ or f(x) = (x+1)/(x–1) for x ≠ 1.

    Suggested tag: cc-9-12m-f-bf-4a

  • (+) Verify by composition that one function is the inverse of another.

    Suggested tag: cc-9-12m-f-bf-4b

  • (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.

    Suggested tag: cc-9-12m-f-bf-4c

  • (+) Produce an invertible function from a non-invertible function by restricting the domain.

    Suggested tag: cc-9-12m-f-bf-4d

• 5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

  Suggested tag: cc-9-12m-f-bf-5