# AII.7h and *AII.7k - Composition of Functions

AII.7    The student will investigate and analyze functions algebraically and graphically. Key concepts include

h)  composition of multiple functions.

Graphing calculators will be used as a tool to assist in investigation of functions.

AII.7  The student will investigate and analyze linear, quadratic, absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic function families algebraically and graphically. Key concepts include

k)  composition of functions, algebraically and graphically.

Bloom's Level:  Analyze

### BIG IDEAS

• I can compare the cost of pizza with number of toppings, find the correlation between number of hours spent in the sun and development of skin cancer, match the amount of money put into a vending machine with the kind of selection that comes out, and equate the growth of plants with amount of food and water they receive.
• I will be able to model various kinds of mathematical relationships and express those relationships in different ways.

### UNDERSTANDING THE STANDARD

• Functions may be used to model real-world situations.
• The domain and range of a function may be restricted algebraically or by the real-world situation modeled by the function.
• A function can be described on an interval as increasing, decreasing, or constant.
• Asymptotes may describe both local and global behavior of functions.
• End behavior describes a function as x approaches positive and negative infinity.
• A zero of a function is a value of x that makes f(x) equal zero.
• If (a, b) is an element of a function, then (b, a) is an element of the inverse of the function.
• Exponential (y = ax) and logarithmic (y = logax) functions are inverses of each other.
• Functions can be combined using composition of functions.

·  Functions may be used to model practical situations.

·  Functions describe the relationship between two variables where each input is paired to a unique output.

·  Function families consist of a parent function and all transformations of the parent function.

·  The domain of a function is the set of all possible values of the independent variable.

·  The range of a function is the set of all possible values of the dependent variable.

·  For each x in the domain of f, x is a member of the input of the function f, f(x) is a member of the output of f, and the ordered pair
(x, f(x)) is a member of f.

·  A function is said to be continuous on an interval if its graph has no jumps or holes in that interval.

·  The domain of a function may be restricted algebraically, graphically, or by the practical situation modeled by a function.

·  Discontinuous domains and ranges include those with removable (holes) and nonremovable (asymptotes) discontinuities.

·  A function can be described on an interval as increasing, decreasing, or constant over a specified interval or over the entire domain of the function.

·  A function, f(x), is increasing over an interval if the values of f(x) consistently increase over the interval as the x values increase.

·  A function, f(x), is decreasing over an interval if the values of f(x) consistently decrease over the interval as the x values increase.

·  A function, f(x), is constant over an interval if the values of f(x) remain constant over the interval as the x values increase.

·  Solutions and intervals may be expressed in different formats, including set notation, using equations and inequalities, or interval notation.  Examples may include:

 Equation/Inequality Set Notation Interval Notation x = 3 {3} x = 3 or x = 5 {3, 5} 0 £ x £ 3 {x|0 £ x £ 3} [0, 3) y ≥ 3 {y: y ≥ 3} [3, ¥) Empty (null) set ∅ { }

·  A function, f, has an absolute maximum located at x = a if f(a) is the largest value of f over its domain.

·  A function, f, has an absolute minimum located at x = a if f(a) is the smallest value of f over its domain.

·  Relative maximum points occur where the function changes from increasing to decreasing.

·  A function, f, has a relative maximum located at x = a over some interval of the domain if f(a) is the largest value of f on the interval.

·  Relative minimum points occur where the function changes from decreasing to increasing.

·  A function, f, has a relative minimum located at x = a over some interval of the domain if f(a) is the smallest value of f on the interval.

·  A value x in the domain of f is an x-intercept or a zero of a function f if and only if f(x) = 0.

·  Given a polynomial function f(x), the following statements are equivalent for any real number, k, such that f(k) = 0:

­   k is a zero of the polynomial function f(x) located at (k, 0);

­  k is a solution or root of the polynomial equation f(x) = 0;

­  the point (k, 0) is an x-intercept for the graph of f(x) = 0; and

­  (xk) is a factor of f(x).

·  Connections between multiple representations (graphs, tables, and equations) of a function can be made.

·  End behavior describes the values of a function as x approaches positive or negative infinity.

·  If (a, b) is an element of a function, then (b, a) is an element of the inverse of the function.

·  The reflection of a function over the line  represents the inverse of the reflected function.

·  A function is invertible if its inverse is also a function. For an inverse of a function to be a function, the domain of the function may need to be restricted.

·  Exponential and logarithmic functions are inverses of each other.

·  Functions can be combined using composition of functions.

Two functions, f(x) and g(x), are inverses of each other if
f(g(x)) = g(f(x)) = x.

### ESSENTIALS

Composition of Functions

All.7h3  Find the composition of two functions.

·  AII.7k1  Determine the composition of two functions algebraically and graphically.

Confirming with Composition

All.7h4  Use composition of functions to verify two functions are inverses.

### KEY VOCABULARY

function, algebraically, graphically, domain, range, limited functions, discontinuous domains, discontinuous ranges, zeros, x-intercept, y-intercept, intervals, increasing, decreasing, asymptotes, end behavior, inverse of a function, composition of multiple functions, restricted domain, restricted range, vertical asymptote, horizontal asymptote, reflections, exponential function, logarithmic function, verify, natural numbers

Updated: Oct 27, 2017