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Properties of Functions:

Definition of a Function: A function is a rule or formula that associates each element in the set X (an input) to exactly one and only one element in the set Y (the output). Different elements in X can have the same output, and not every element in Y has to be an output.

Definition of the Domain of a Function: The set of all possible inputs of a function is defined as the domain. The domain of a real-valued function defined by a formula for y in terms of x will be the set of all x input-values that result in a real y output-value unless the domain of the function is further restricted.

Definition of the Range of a Function: The set of all possible outputs of a function is defined as the range. The range of a real-valued function defined by a formula for y in terms of x will be the set of all y output-values that result from the x input-values in the domain.

Function Notation: Given that f(x) is given by some formula containing x, f(B) will be the same formula with each x replaced by B.

Linear Function Definition: If a function may be written in the form f(x) = mx + b where x is the independent variables and m and b are constants, then f(x) represents a linear function. The variable m is defined as the slope and the point (0, b) represents the y-intercept. An equation in this form is known to be in Slope-Intercept Form.

Linear Function Slope Definition: Given that f(x) = mx + b, then m is defined as the slope where:
 
for any two points (x1, y1) and (x2, y2) on the line.  Graphically, the slope represents the change in y with respect to x on the graph of the line.

Linear Functions of Parallel Lines If two linear functions are given by f(x) = m1x + b and g(x) = m2x + b, and m1 = m2, then the graphs of f(x) and g(x) will consist of two lines that are parallel to each other.

Linear Functions of Perpendicular Lines If two linear functions are given by f(x) = m1x + b and g(x) = m2x + b, and m1 = -1/m2, then the graphs of f(x) and g(x) will consist of two lines that are parallel to each other.

Graphs of Even Functions Given a function f(x), if f(c) = f(-c) for all c in the domain, then f(x) is an even function and its graph will have symmetry with respect to the y-axis.

Graphs of Odd Functions Given a function f(x), if f(c) = -f(-c) for all c in the domain, then f(x) is called an odd function and its graph will have symmetry with respect to the origin.  Symmetry with respect to the origin implies that a 180 degree rotation of the graph about (0,0) results in an identical graph.

Functions Shifted Left Given a function f(x) and its graph and a value of c>0, the graph of f(x + c) will be a shift of the graph of f(x) left by "c" units. This is known as the Left Shift Function Rule.

Functions Shifted Right Given a function f(x) and its graph and a value of c>0, the graph of f(x - c) will be a shift of the graph of f(x) right by "c" units.  This is known as the Right Shift Function Rule.

Functions Shifted Up Given a function f(x) and its graph and a value of c>0, the graph of f(x) + c  will be a shift of the graph of f(x) up by "c" units. This is known as the Vertical Shift Function Rule.

Functions Shifted Down Given a function f(x) and its graph and a value of c>0,  the graph of f(x) - c  will be a shift of the graph of f(x) down by "c" units. This is known as the Vertical Shift Function Rule.

Function Reflected Across X-axis Given a function f(x) and its graph, the graph of g(x) = -f(x)  will be a reflection of the graph of f(x) across the x-axis.  This is known as the X-axis Reflection Function Rule.

Function Reflected Across Y-axis Given a function f(x) and its graph, the graph of g(x) = f(-x)  will be a reflection of the graph of f(x) across the x-axis.  This is known as the Y-axis Reflection Function Rule.

Function Vertically Stretched Or Shrunk Given a function f(x) and its graph and a value of c>0, the graph of g(x) = c●f(x) will be a vertical stretch of the graph of f(x).  This means that all y-values of g(x) will be equal to c times the respective y-values of f(x). This is known as the Vertical Stretch Function Rule.

Definition of a Polynomial Function If f(x) may be written in the form a1xn + a2xn-1 + a3xn-2 + .... + an, then f(x) is a polynomial function of degree n  where a1, a2, ... an are real coefficients.  Linear functions are 1st degree polynomials and quadratic functions are 2nd degree polynomials.

Graphs of Polynomials Given a function f(x) is a polynomial, it's x-intercepts will be located at the x-values x=c such that f(c) = 0.  Other solution points on the graph will be located between each two x-intercepts.

Standard Form of Quadratic Functions Quadratic functions of the form f(x) = ax2 + bx + c may always be rewritten in the form  y = a(x - h)2 + k.  Function shift rules may then be applied to state that the graph will be a vertical stretch of y = x2 and will be shifted right, left, up, or down according to the values of h and k.

Graphs of Quadratic Functions in Form f(x) = ax2 + bx + c Given f(x) = ax2 + bx + c, the graph will be a shift of g(x) = ax2 (meaning it has the same shape), and will have a vertex at x=-b/2a, y = f(-b/2a).

Property of The Vertex of a Quadratic Function The vertex of f(x) = ax2 + bx + c will be the lowest point of the graph if a>0 and will be the highest point of the graph if a<0.  The vertex represents the minimum value of the function for a>0 and represents the maximum value of the function if a<0.

Function Operations Given two functions f(x) and g(x), the operations (f+g)(x), (f-g)(x), (fg)(x), and (f/g)(x) are defined in the following way:

  • (f + g)(x) = f(x) + g(x)  and is sometimes denoted f+g
  • (f - g)(x) = f(x) - g(x)  and is sometimes denoted f - g
  • (fg)(x) = f(x)● g(x)  and is sometimes denoted fg
  • (f/g)(x) = f(x)/g(x)  provided g(x)0.    This is sometimes denoted f/g

Function Composition Given two functions f(x) and g(x), the function compostion (fog)(x), is defined in the following way:

(f o g)(x) = f [g(x)]  and is sometimes denoted as f o g

In essence, composition implies that you input the entire formula of the second function in for each x-value of the the formula in the first function, assuming x is the variable used.

Definition of Inverse Functions Given two functions f(x) and g(x), if (f o g)(x) = x and (g o f)(x) = x, then f(x) is the inverse of g(x) and g(x) is the inverse of f(x). Each of these functions reverses the operations of the other function in reverse order.  In that sense, the inverse of f(x) will consist of the identical formula with x and y interchanged - the solution for y results in "reversing" all operations on x and thus results in the formula for the inverse function.

We denote the inverse of f(x) as f -1(x) and we denote the inverse of g(x) as g-1(x).

Domain and Range of Functions That Are Inverses of Each Other Given two functions f(x) and g(x) are inverses of each other, then

The Domain of f(x) will consist of the same interval as the Range of g(x).

The Range of f(x) will consist of the same interval as the Domain of g(x).

One-To-One Requirement For f(x) To Have an Inverse Function Given a function f(x), it will only have an inverse if and only if each y-value in it's range corresponds to only 1 x-value in it's specified domain.  When this is the case that each y is obtained from only 1 x-value we say f(x) is a one-to-one function.

Note that a graphical way to determine that f(x) is not one-to-one is to show that a horizontal line passes through more than 1 point. This is often referred to as the Horizontal Line Test.

 

 

 

 

 

 

 

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