Chapter 3: Functions
function
A function is a relation in which each possible input value leads to exactly one output value. We say "the output is a function of the input.
g(x)=f(x−h)
Horizontal Shift to the right
What is the symbol notation for a composite function? How does this read?
(f∘g)(x)=f(g(x)) We read the left-hand side as "f composed with g at x," and the right-hand side as"f of g of x."
identity function - Equation -Graph
- Equation: f(x)=x -Graph: See picture
HOW TO Given the formula for a function, determine if the function is even, odd, or neither.
1. Determine whether the function satisfies f(x)=f(−x). If it does, it is even. 2. Determine whether the function satisfies f(x)=−f(−x). If it does, it is odd. 3. If the function does not satisfy either rule, it is neither even nor odd.
HOW TO Given a function composition f(g(x)) determine its domain.
1. Find the domain of g. 2. Find the domain of f. 3. Find those inputs x in the domain of g for which g(x) is in the domain of f. That is, exclude those inputs x from the domain of g for which g(x) is not in the domain of f. The resulting set is the domain of f°g.
HOW TO Given a function represented by a formula, find the inverse.
1. Make sure f is a one-to-one function. 2. Solve for x. 3. Interchange x and y.
HOW TO Given a function, reflect the graph both vertically and horizontally.
1. Multiply all outputs by -1 for a vertical reflection. The new graph is a reflection of the original graph about the x-axis. 2. Multiply all inputs by -1 for a horizontal reflection. The new graph is a reflection of the original graph about the y-axis.
even function
A function is called an even function if for every input x, f(x)=f(−x). In short, the function will possess the remain output sign regardless of the input value sign.
odd function
A function is called an odd function if for every input x if f(x)=−f(−x) The graph of an odd function is symmetric about the origin.
one-to-one function
A one-to-one function is a function in which each output value corresponds to exactly one input value. There are no repeated x- or y-values.
What type of function do we use to define absolute value?
A piecewise function
Piecewise function
A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains.
What does a rate of change describe
A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are "output units per input units."
rate of change
A ratio between the change in one variable and the change in another. Same as "slope"
What in a function changes to create a horizontal shift of a graph?
A shift to the input
What is the purpose of shifting functions ?
Altering functions allows us to model real world problems
How do we create a vertical shift?
By adding a positive or negative constant.
average rate of change
Change in y over change in x; make a table using your graphing calculator to help!
output
Each value in the range
Constant function - Equation -Graph
Equation: f(x)=c, where c is a constant
What is function composition?
Function composition is only one way to combine existing functions. The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions.
vertical stretch or vertical compression
Given a function f(x), a new function g(x)=af(x), where a is a constant, is a vertical stretch or vertical compression of the function f(x). If a>1, then the graph will be stretched. If 0<a<1, then the graph will be compressed. If a<0, then there will be combination of a vertical stretch or compression with a vertical reflection.
horizontal stretch or horizontal compression
Given a function f(x), a new function g(x)=f(bx), where b is a constant, is a horizontal stretch or horizontal compression of the function f(x). If b>1, then the graph will be compressed by 1b. If 0<b<1, then the graph will be stretched by1b. 1b. If b<0, then there will be combination of a horizontal stretch or compression with a horizontal reflection.
What changes to a function occur to result in a vertical reflection?
Given a function f(x), a new function g(x)=−f(x) is a vertical reflection of the function f(x), sometimes called a reflection about (or over, or through) the x-axis. In short... The output is negative
How do we represent that functions are inverse of one another symbolically?
Given a function f(x),we represent its inverse as f⁻¹(x), read as "f inverse of x." The raised −1 is part of the notation. It is not an exponent; it does not imply a power of−1 .
What changes to a function occur to result in a horizontal reflection?
Given a function, f(x), a new function g(x)=f(−x) is a horizontal reflection of the function f(x), sometimes called a reflection about the y-axis. In short, the original function's input (x component) becomes negative
g(x)=f(x+h)
Horizontal shift to the left
What type of functions does defines the following example: A reversible heat pump is a climate-control system that is an air conditioner and a heater in a single device. Operated in one direction, it pumps heat out of a house to provide cooling. Operating in reverse, it pumps heat into the building from the outside, even in cool weather, to provide heating. As a heater, a heat pump is several times more efficient than conventional electrical resistance heating.
Inverse functions; where one function is the opposite of the other
Did adding or subtracting values to the input or output values affect the direct shape of the graph?
No, it merely shifted them.
Do all functions have inverses?
No, not all functions have inverses.
Is it possible for a function to have more than one inverse?
No. If two supposedly different functions, say g and h, both meet the definition of being inverses of another function f, then you can prove that g=h. We have just seen that some functions only have inverses if we restrict the domain of the original function. In these cases, there may be more than one way to restrict the domain, leading to different inverses. However, on any one domain, the original function still has only one unique inverse.
What do even and odd characteristics determine about a function?
Some functions exhibit symmetry so that reflections result in the original graph. If a function is even or odd it will present symmetry
Provide real world examples of function composition.
Suppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year. Notice how we have just defined two relationships: The cost depends on the temperature, and the temperature depends on the day. One example of a composition of two functions could be a vehicle's acceleration. The vehicle's speed is a function of time, of F(t), while it's acceleration is a function of its speed. Therefore, G(F(t))=acceleration. As such, acceleration is basically a function of a function, or a function of the function that determines speed. Interestingly enough, acceleration is also the derivative of the velocity function.
absolute maximum
The absolute maximum off f at x=c x=c is f(c) f(c) where f(c)≥f(x) f(c)≥f(x) for all x x in the domain off.
What is the average rate of change?
The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.
What is often referred to in terms of absolute value?
The distance between two objects
What is the domain of a composite function?
The domain of the composite function f∘g is all x x such that x is in the domain of g and g(x) is in the domain of f.
Inverse Function
The function that results from exchanging the domain (x-values) and range (y-values) of a one-to-one function.
What is the relationship of the original function and the inverse function when you graph them?
The graph of f⁻¹ is the graph of f(x) reflected about the diagonal line y=x, which we will call the identity line.
FUNCTION NOTATION
The notation y=f(x) defines a function named f. This is read as "y" is a function of x. The letter x represents the input value, or independent variable. The letter y, or f(x), f(x), represents the output value, or dependent variable.
dependent variable
The outcome factor; the variable that may change in response to manipulations of the independent variable.
The outputs of _____________ are the inputs of _____________.
The outputs of the function f are the inputs to f⁻¹, so the range off f is also the domain of f⁻¹.
What is the domain of the inverse function?
The range of the original function, f, is the domain of the inverse, because the outputs of the function f (the range) are the inputs ( domain) of the inverse function.
composite function
The resulting function is known as a composite function
domain of a function
The set of all possible input values for the function
range of a function
The set of output values of a function.
Intuitively, it seems we would shift to the right for when adding a constant to the input (x- value) and shift to the left when subtracting a constant from the input. Then why is it when we add a value, we shift left and subtract a value we shift right?
The thought process should be as follows: when adding or subtracting the input, we want the output to stay the same. For example, if f(x)=x², then g(x)=(x−2)² is a new function. Each input is reduced by 2 prior to squaring the function. The result is that the graph is shifted 2 units to the right, because we would need to increase the prior input by 2 units to yield the same output value as given in f. In short, we do the opposite to keep the output value the same as it was prior to adding a constant to the input. We want the outputs to stay the same value.
Vertical Line Test
The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value.
Why is the inverse function a reflection of the original function y=x?
This relationship will be observed for all one-to-one functions, because it is a result of the function and its inverse swapping inputs and outputs. This is equivalent to interchanging the roles of the vertical and horizontal axes.
Solving an Absolute Value Equation
To solve an equation such as, 8=| 2x−6 |, we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.
If a function is not one to one, how can we create an inverse function?
We can still restrict the function to a part of its domain on which it is one-to-one.
When do we use piecewise functions; give an example.
We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain "boundaries." Examples: We often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions.
What can we do if a function does not have an inverse?
When a function has no inverse function, it is possible to create a new function where that new function on a limited domain does have an inverse function. For example, the inverse of f(x)=√x is f⁻¹(x)=x², because a square undoes a square root; but the square is only the inverse of the square root on the domain [0,∞), since that is the range of f(x)=√x.
What does f(x) mean?
Y
Is there a method to change the shape of the original graph? How?
Yes, we can compress or stretch a graph by multiplying the inputs or outputs by some quantity. We can transform the input values or the output values of the graph by multiplying them by a value; each action possesses different effects on the graph.
Can you combine horizontal and vertical shifts? Provide an example.
Yes, you can. Example: y= |x+1| - 3 This is interpreted as follows: Vertical shift upward of 3 Horizontal shift of one to the left. Original Function: y=|x|
set-builder notation
a notation for a set that uses a rule to describe the properties of the elements of the set; example {x | x ≥ 3 and x ∈ ℵ}
relation
a set of ordered pairs
independent variable
a variable (often denoted by x ) whose variation does not depend on that of another.
horizontal line test
an easy way to determine if it is a one-to-one function is to use the horizontal line test
input
each value in the domain is also known as an input value
Quadratic Function
f(x) = x²
Reciprocal Function
f(x)=1/x
Reciprocal squared Function
f(x)=1/x²
Cubic Function
f(x)=x³
Finding the Zeros of an Absolute Value Function
f(x)=|4x+1|−7, find the values of x such that f(x)=0.
Absolute value function
f(x)=|x|
Cube root
f(x)=³√x
Square root function
f(x)=≡√x
VERTICAL SHIFT
f(x)±k The graph moves k parallel to the y-axis
increasing function
for a function f and any number x₁ and x₂ in the domain of f, the function f is increasing over an open interval if for every x₁<x₂ in the interval f(x₁)<f(x₂)
Decreasing function
for a function f and any numbers x₁ and x₂ in the domain of f, the function f is decreasing over an open interval if for every x₁>x₂ in the interval f(x₁)>f(x₂)
local maximum
for a function f, f(a) is a local max if there is an interval around a such that f(a)>f(x) for all values of x in the interval where x≠a
local minimum
for a function f, f(a) is a local min if there is an interval around a such that f(a)<f(x) for all values of x in the interval where x≠a
What are odd functions symmetric about?
origin
Horizontal Reflection
reflects a graph horizontally over the y - axis
vertical reflection
reflects a graph vertically across the x-axis
What is a vertical shift?
shifting the function up or down on the graph, in the y direction.
The Greek letterΔ (delta) represents what?
signifies the change in a quantity
Define inverse function
which is a function for which the input of the original function becomes the output of the inverse function and the output of the original function becomes the input of the inverse function.
What are even functions symmetric about?
y- axis