Understanding Functions

Functions

• Functions describe the dependence of one quantity on another
• Examples of functions: consumer demand based on product price, water temperature relative to heating time

Function Definition and Notation

• The term function indicates a correspondence where each input (x) has exactly one output (y)
• Functions can be named f or g
• f(x) denotes the output corresponding to the input x
• The input x enters the function and produces an output y = f(x)

Function Components

• Domain: represents all possible inputs for a function
• Range: represents all possible outputs of a function

Representing Functions

• Arrow diagrams can show the correspondence of inputs to outputs
• Equations with two variables may represent y as a function of x if, for every x, there is only one y

Domain Determination

• The domain of a function is the set of all x-values so that the corresponding y-value is a real number

Types of Functions and Domain Restrictions

Polynomials

• The domain is all real numbers

Fractions

• Values of x that make the denominator equal to zero are excluded from the domain

Radicals

• With odd indices, the domain covers all real numbers if the radicand is a polynomial
• With even indices, domain values require the radicand to be non-negative (≥ 0)

Evaluating Functions

• Evaluating a function f means determining the value of f(x) for a specific x-value
• The given value should be substituted for x, and then simplify

Function Actions

• One acts on the input x to produce the output f(x)

Determining if Functions are Equal

• It can be determined values that make two functions equal
• Set them equal to each other f(x) = g(x)
• Transfer all terms to one side
• Factor, find the values of x

Zero of a Function

• A zero of a function is an x-value which results in an output of zero
• Set f(a) = 0
• Find inputs to get outputs equal to 0

Graphs

Cartesian Plane

• A plane is formed by two real number lines intersecting at right angles
• x-axis: horizontal number line.
• y-axis: vertical real number line.
• origin: where the two axes intersect and where x and y are equal to zero (0,0)
• quadrants are the four parts of the plane

Coordinates

• Each point corresponds with an ordered pair (x, y) of real numbers x and y
• They are called the coordinates of the point

Table of Values

• The graph of an equation is defined by the set of all points that are solutions to the equation
• One way to graph an equation is to generate a set of ordered pairs that correspond to the equation

Intercepts

• A point at which a graph meets the x-axis is called an x-intercept
• A point at which a graph meets the y-axis is called a y-intercept
• An x-intercept has the form (x, 0)
• A y-intercept has the form (0, y)

Vertical Line Test

• This determines if a graph represents a function
• A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point

Evaluating functions

• A function can be evaluated using its graph
• Locate the point on the graph corresponding to the x-value and determine the y-value of the point

Domain and Range from a Graph

• The domain is the set of x values for which f(x) is defined as a real number
• The range represents the y values

Using a Graph

• Read the graph from left to right from the domain
• For the range, read the graph from bottom to top

Identifying Intervals on Graphs

• Graphs can be increasing, decreasing, or constant
• It is indicated if y-values of the points get larger or smaller
• This is determined by reading from left to right

Intervals

• Intervals are always listed in terms of x values, and the endpoints are not included.

Relative Maximum

• When a function switches from increasing to decreasing

Relative Minimum

• When a function changes from decreasing to increasing

Intervals with Positive or Negative Functions

• Intervals can be identified by stating where the graph sits above the x-axis
• Functions are negative or below the x-axis
• Intervals are listed in terms of the x-values

Transformations of Common Functions

Term

• c represents a positive real number

Horizontal Shifts

• h(x) = f(x – c) produces a horizontal shift of c units to the right
• h(x) = f(x + c) produces a horizontal shift of c units to the left
• To note, the real number is added or subtracted inside the common function.

Vertical Shifts

• h(x) = f(x) − c produces a vertical shift of c units downward
• h(x) = f(x) + c produces a vertical shift of c units upward
• The real number is added or subtracted outside the common function.

Reflections

• h(x) = f(–x) indicates reflection about the y-axis
• Negative one is multiplied inside the common function
• h(x) = –f(x) indicates reflection about the x-axis, where −1 is multiplied outside the common function

Transformations Order

• Following order is applicable for more than one transformation
• (horizontal, reflect, vertical - HRV)

Graphing Polynomial Functions

• This requires knowledge of points, end behavior, and intercepts

Turning Points - Graphs

• A polynomial function of degree n has at most n real zeros
• It has n-1 turning points (relative extrema)

Coefficient Test

• Can use a leading coefficient to test to determine the end behavior of the graph

Even Degree

• Compare to y = x^2, both ends go up, and compare y = -x^2; both ends go down

Odd Degree

• Compare to y = x^3 - Left goes down, right goes up
• Compare to y = -x^3 - Left goes up, right goes down

Roots

• The zeros of a polynomial indicate where the graph has x-intercepts
• Factor to solve
• If x = a is a zero, (x – a) is a factor

Multiplicity with factoring

• When factored, some polynomials have repeated factors
• The number of times a factor is repeated is referred to as the multiplicity of the zero

Multiplicity of K

• In general x – a)^k^ yields a repeated zero of multiplicity k
• If k is odd, the graph crosses the x-axis at x = a
• If k is even, the graph touches the x-axis at x = a

Algebra of Functions

• Let f and g be two functions with overlapping domains
• Arithmetic combinations of f and g are defined for all x-values common to both domains

Sum:

• (f + g)(x) = f(x) + g(x)

Difference:

• (f − g)(x) = f(x) − g(x)

Product:

• (fg)(x) = f(x) · g(x)

Quotient:

• (f/g)(x) = f(x) / g(x), (assuming that g(x) ≠ 0)

Function Composition

• Another way to combine two functions; when two functions are composed, one function becomes the input for the other
• Composition of the function f with g is (f ◦ g)(x) = f(g(x))
• The inner function, which becomes the input, g, for the outer function f

Function Decomposition Terms

• Use of inside and outside
• Be aware of x and inner values