Building Functions

Build a function that models a relationship between two quantities

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

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

         b. 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.

         c. (+) 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.

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

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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.
4. Find inverse functions.

         a. 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.

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

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

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

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

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