Algebra II End-of-Course Exam Content Standards—Core: Function Operations and Inverses (Priority: 15%)
Successful students will be able to perform function operations of addition, subtraction, multiplication, division, and composition and to combine several functions defined over restricted domains to form a piecewise-defined function. They will be able to determine, graph and analyze the inverse of a function and use composition to determine whether two functions are inverses. There are a variety of types of test item including some that cut across the objectives in this standard and require students to make connections.
F1. Function operations
a. Combine functions by addition, subtraction, multiplication, and division.
Example: If f(x) = 3x4 - 5x3 + 3, and g(x) = x4 - 3x3 + 2x2 + 5, then f(x) - g(x) -
(3x4 - 5x3 + 3) - (x4 - 3x3 + 2x2 + 5)
= (3x4 - x4) + (-5x3 - (-3x3)) - 2x2 + (3 - 5)
= 2x4 - 2x3 - 2x2 - 2
Example: If h(x) - x4 - 16 and j(x) - x - 2, then h(x) ÷ j(x) - (x4 - 16) ÷ (x - 2) - 
- (x + 2)(x2 + 4) for x - 2.
Example: If r(x) = x2 + 3x - 1 and s(x) = x + 3, then r(x) · s(x) = (x2 + 3x - 1)(x + 3) = x3 + 6x2 + 8x - 3
b. Determine the composition of two functions, including any necessary restrictions on the domain.
Example: If f(x) - 3x - 2, and 
find f(g(x)) and f(f(x)). Include domain restrictions on each.
Example: If 
, and 
, determine whether or not f and g are inverses and explain how you know.
F2. Inverse functions
a. Describe the conditions under which an inverse relation is a function.
- Consider graphic conditions for an inverse relation to be a function.
- Use the horizontal line test to determine whether the inverse of a function is also a function. Consider domain restrictions for existence of an inverse function.
- Recognize that the inverse of a quadratic function is a function only when its domain is restricted.
b. Determine and graph the inverse relation of a function.
- Include inverses which may not be functions.
Example: The inverse relation of y - x2 is y -
. Explain why an inverse function would be only either the positive or the negative part of the graph. - Explain why the graphs of a function and its inverse are reflections of each other over the line y = x.
- Show that when the inverse of a function is a function f-1(f(x)) = x and f(f-1(x)) - x
Example: Determine the inverse relation for f(x) = 3x2 + 5.
Example: Determine g(x) when g-1(x) = 9x - 81.
- Inverses of exponential functions may be required graphically but not in algebraic form.
Example: Graphically determine f-1(x) when f(x) = 2x - 7.
F3. Piecewise-defined functions
a. Determine key characteristics of absolute value, step, and other piecewise-defined functions.
- Key characteristics include vertices, intercepts, end behavior, slope of linear sections, and discontinuities.
- Determine the vertex, slope of each branch, intercepts, and end behavior of an absolute value graph.
Tasks related to this benchmark: Function Transformations
b. Represent piecewise-defined functions using tables, graphs, verbal statements, and equations. Translate among these representations.
- Interpret the algebraic representation of a piecewise defined function; graph it over the appropriate domain.
- Write an algebraic representation for a given piecewise defined function.
Tasks related to this benchmark: Function Transformations
c. Recognize, express, and solve problems that can be modeled using absolute value, step, and other piecewise-defined functions. Interpret their solutions in terms of the context.
- This includes using and interpreting appropriate units of measurement and precision for the given application.
- Applications may include postage rates, salary increases, or cellular telephone charges.