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.

Example: If *f(x)* = 3*x*^{4} - 5*x*^{3} + 3, and *g(x) = x*^{4} - 3*x*^{3} + 2*x*^{2} + 5, then *f(x) - g(x)* -
(3*x*^{4} - 5*x*^{3} + 3) - (*x*^{4} - 3*x*^{3} + 2*x*^{2} + 5)

= (3*x*^{4} - *x*^{4}) + (-5*x*^{3} - (-3*x*^{3})) - 2*x*^{2} + (3 - 5)

= 2*x*^{4} - 2*x*^{3} - 2*x*^{2} - 2

Example: If *h(x) - x*^{4} - 16 and *j(x) - x* - 2, then *h(x) ÷ j(x)* - (*x*^{4} - 16) ÷ (*x* - 2) - - (*x* + 2)(*x*^{2} + 4) for *x* - 2.

Example: If *r(x) = x*^{2} + 3*x* - 1 and *s(x) = x* + 3, then *r(x) · s(x)* = (*x*^{2} + 3*x* - 1)(*x* + 3) = *x*^{3} + 6*x*^{2} + 8*x* - 3

Example: If *f(x)* - 3*x* - 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.

- 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.

- Include inverses which may not be functions.
Example: The inverse relation of

*y - x*^{2}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)) - xExample: Determine the inverse relation for

*f(x)*= 3*x*^{2}+ 5.Example: Determine

*g(x)*when*g*^{-1}(*x*) = 9*x*- 81. - Inverses of exponential functions may be required graphically but not in algebraic form.
Example: Graphically determine

*f*when^{-1}(x)*f(x)*= 2^{x}- 7.

- 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

- 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

- 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.