Introduction to functions
Review of the cartesian product and definition of a function
Cartesian product
The cartesian product of the real numbers is the collection of all tuples (x,y) such that both x and y are real numbers.
The coordinates (2{,}3), (7,-\pi), and (0{,}0) are all elements of the cartesian product of the real numbers.
The xy-plane
A xy-plane is a visual of the cartesian product of the real numbers. It is made of two real number lines intersected perpendicularly : the two dimensional plane is determined by these lines :
- The horizontal position of a point determines the value of the first coordinate.
- The vertical position of a point determines the value of the second coordinate.
The horizontal axis is referred to as the x-axis and the vertical axis is referred to as the y-axis.
Function
A function is a relation of inputs and outputs which assigns a unique output to each input. Functions are typically denoted by f(x). The variable x is a placeholder for the inputs and the f is the function name. However, we can also use other letters such as g(x) or h(x) to represent a function.
Consider the following function:
f(x)=x^2
It assigns to each real number the real number obtained by squaring it. For example, the function f(x) assigns the real number 3 to the real number 9 since:
f(3)=3^2=9
It is common to use y to denote a function. For example, we might often write y=x^2 instead of f(x)=x^2.
The graph of a function f\left(x\right) is the set of all points of the form \left(x,f\left(x\right)\right) in the xy -plane.
The following is the graph of the function f\left(x\right)=x^2. Observe that the point \left(3{,}9\right) is on the graph because f\left(3\right)=3^2=9.
Domain and range of a function
Domain of a function
The domain of a function f(x) is the collection of real numbers which can be inputted into f(x).
Consider the following function:
f(x)=x^2
It has the domain of all real numbers since any real number can be plugged into x^2.
Consider the following function:
f(x)=\dfrac{1}{x}
It has the domain of all real numbers except 0 since we can divide by all real numbers except for 0.
Range of a function
The range of a function f(x) is the collection of real numbers which are outputs of f(x).
Consider the following function;
f(x)=x^2
Its range is [0,\infty) since the square of any real number is nonnegative and every positive number is the square of a real number.
Consider the following function:
f(x)=x-1
Its range is all real numbers. This is because if a is a real number then:
f(a+1)=a+1-1=a
Graph of a function
The graph of a function f(x) is the collection of all point (a,f(a)) on the xy-plane such that a is in the domain of f(x).
The graph of the function f(x)=x^2 is a parabola.
A function can have only one output for every input. Therefore the graph of a function must pass the vertical line test. This means that given the graph of a function, a vertical line can only intersect the graph of the function one time at most.
The graph above cannot be the graph of a function since the dashed vertical line intersects the graph in more than one place.
Given the graph of a function f(x) we can find the domain and the range of that function.
- The domain is all values on the x-axis for which the graph either lies above or below.
- The range is all values on the y-axis for which the graph either lies to left or right of.
In the following graph the domain is highlighted in blue on the x -axis and the range is highlighted in red on the y -axis.
The domain of the function is [-2{,}1). The closed circle at the left endpoint indicates that -2 is in the domain whereas the open circle at the right endpoint indicates 1 is not in the domain.
Similarly, the range is then [0{,}4].
First properties of functions and operations
We can add, subtract, multiply and divide by functions.
Addition of functions
If f(x) and g(x) are functions then the function (f+g)(x) is defined by:
(f+g)(x)=f(x)+g(x)
Consider the following functions:
- f(x)=x^2-x^3
- g(x)=2x^3+7
(f+g)(x) is the following function:
(f+g)(x)=x^2-x^3+2x^3+7=x^2+x^3+7
The graph of the sum of two functions f(x) and g(x) is found by summing the points on the two functions piecewise.
For example, in the following graph the graph of f(x) is in red, the graph of g(x) is in blue, and the graph of (f+g)(x) is in black.
Subtraction of functions
If f(x) and g(x) are functions then the function (f-g)(x) is defined by:
(f-g)(x)=f(x)-g(x)
Consider the following functions:
- f(x)=x^2-x^3
- g(x)=2x^3+7
(f-g)(x) is the following function:
(f+g)(x)=x^2-x^3-(2x^3+7)=x^2-3x^3-7
Similar to addition, the graph of the difference of two functions f(x) and g(x) is found by subtracting the points on the two functions piecewise.
For example, in the following graph the graph of f(x) is in red, the graph of g(x) is in blue, and the graph of (f-g)(x) is in black.
Multiplication of functions
If f(x) and g(x) are functions then the function (f g)(x) is defined by:
(f g)(x)=f(x)g(x)
If:
f(x)=x^2-x^3
And:
g(x)=2x^3+7
Then:
(fg)(x)=(x^2-x^3)(2x^3+7)=2x^5+7x^2-2x^6-7x^3
Division of functions
If f(x) and g(x) are functions then the function \dfrac{f}{g}(x) is defined for any real x such that g\left(x\right)\neq0 by:
\dfrac{f}{g}(x)=\dfrac{f(x}{g(x)}
Consider the following functions:
- f(x)=x^2-x^3
- g(x)=2x^3+7
\dfrac{f}{g}(x) is the following function:
\(\displaystyle\frac{f}{g}(x)= \frac{x^2-x^3}{2x^3+7
Basic concepts
Average rate of change of a function
The slope of the line connecting two points
The slope of the line connecting two points (x_0,y_0) and (x_1,y_1) in the xy-plane is:
\dfrac{y_1-y_0}{x_1-x_0}
The following is the graph of a line which contains the points \left(0{,}1\right) and \left(2{,}5\right).
Therefore the slope of the line is:
\dfrac{5-1}{2-0}=\dfrac{4}{2}=2
Suppose f(x) is a function and two numbers x_0 and x_1 in the domain of f(x). The points (x_0,f(x_0)) and (x_1, f(x_1)) are on the graph of f(x) and we can compute the rate of change of f(x) from these two points using the formula for slope.
Average rate of change
Let f(x) be a function and x_0 and x_1 two numbers in the domain of f(x). The average rate of change of the function f(x) from x_0 to x_1 is:
\dfrac{f(x_1)-f(x_0)}{x_1-x_0}
Consider de the function :
f(x)=x^2
Its average rate of change from 2 to 4 is:
\dfrac{4^2-2^2}{4-2\\}=\dfrac{16-4}{2}=\dfrac{12}{2}=6
The graph of a function f(x) is given below. The slope of the line connecting the labeled points on the graph is the average rate of change of f(x) from x_0 to x_1.
The average rate of change of a function f\left(x\right) over an interval \left[a,b\right] measures how much on average a function has changed over the interval.
The x-intercepts and y-intercepts of a function
y-intercept
The y-intercept of a function f(x) is the coordinate of where the function crosses the y-axis. The y-intercept of the graph of f(x) is:
(0,f(0))
Consider the following function:
f(x)=x^2+1
It has y-intercept at:
(0,f(0))=(0{,}1)
x-intercept
The x-intercepts of a function f(x) are the coordinates where the graph of f(x) touches the x-axis. The graph of a function touches the x-axis at a point x_0 only if f(x_0)=0. So to find the x-intercepts of a function we need to find all solutions to the equation f(x)=0.
Consider the following function:
f(x)=x-1
It has one x-intercept of (1{,}0) since x-1=0 if and only if x=1.
Consider the following function:
f(x)=(x-3)(x-1)
It has x-intercepts at (3{,}0) and (1{,}0) since:
(x-3)(x-1)=0
if and only if:
x-1=0 or x-3=0
x=1 or x=3
The graph of a function f(x) is given below. The points labeled A,B,C are the x-intercepts and the point labeled D is the y-intercept.
Not all functions have x -intercepts or y -intercepts:
- A function f\left(x\right) has a y -intercept if and only if 0 is in the domain of f\left(x\right).
- A function f\left(x\right) has a x -intercept if and only if 0 is in the range of f\left(x\right).
Increasing or decreasing behavior and extrema
Increasing
A function is increasing at a point on a graph if the y-values of the graph increase as the x-values increase.
Consider the following function:
f(x)=2x
It is increasing everywhere because as x becomes larger the corresponding y-values also increase.
Decreasing
A function is decreasing at a point on a graph if the y-values are decreasing as the x-values increase.
Consider the following function:
f(x)=-2x
It is decreasing everywhere because the y-values decrease as x-values increase.
The graph of function allows one to visually observe where the function is increasing and decreasing. If we imagine a particle traveling on the graph of a function from left to right then the function is increasing whenever the object moves upwards and the function is decreasing whenever the particle is moving downwards.
In the following graph, the blue areas represent the portions of the graph where the function is increasing and the red areas represent the portions of the graph where the function is decreasing.
The intervals in which the function is increasing are (-2, -1) and (1{,}2). The interval where the function is decreasing is (-1{,}1).
Extrema
The local extrema of a function are the points on the graph of a function where a graph goes from increasing to decreasing or decreasing to increasing. In the case that the graph has end points and is either increasing or decreasing towards those endpoints, then we include these points as local extrema.
A global extrema is any local extrema where the corresponding y -value is either an absolute maximum or minimum of the function.
In the previous graph the coordinates (-2{,}0), (-1{,}3), (1,-3), and (2{,}0) are the local extremal points.
The absolute extremal points are \left(-1{,}3\right) and \left(1,-3\right).
Composition and inverse of functions
Composition of functions
Suppose f(x) and g(x) are functions. The composition of f(x) and g(x) is the function (f\circ g)(x) defined by:
(f\circ g)(x)=f(g(x))
Consider the following functions:
- f(x)=x^2
- g(x)=x+1
We have:
(f\circ g)(x)=(x+1)^2=x^2+2x+1
And:
(g\circ f)(x)=x^2+1
Inverse of a function
Let f(x) be a function. We say that g(x) is the inverse of f(x) if :
- (f\circ g)(x)=x
- (g\circ f)(x)=x
To find the inverse of a function y=f(x) we switch y and x and then solve for y.
Consider the following function:
y=x+1
The inverse of the function is found as follows:
x=y+1
x-1=y
y=x-1
Consider the following function:
y=\dfrac{x+1}{x}
To find its inverse we solve the equation for y. The equation is solved as follows:
x=\dfrac{y+1}{y}=1+\dfrac{1}{y}
x-1=\dfrac{1}{y}
y=\dfrac{1}{x-1}
Not every function has an inverse. The graph of a function will have to pass the horizontal line test in order to have an inverse. This means that given the graph of the function will have an inverse only if every horizontal line intersects the graph in no more than one place.
For example, the function f(x)=x^2 does not have an inverse because it fails the horizontal line test.
The graph of the inverse of a function is obtained by reflecting the original graph across the line y=x.
