Unit 1: Functions

Jonny Flieder and George Laveran

To Incoming Students...

Although you may have a basic knowledge of functions from your time in Algebra II, this page will reinforce the basics of functions and strengthen your overall conceptual understanding of the topic.

Topics

Intro to Functions

Types of Functions

Transformations

Composition of Functions

Inverse Functions

Special Transformations

Unit 1-1: Functions

What is a function?

By definition, a function is a set of ordered pairs for which each value of the independent variable in the domain has only one corresponding value of the dependent variable in the range.

Rather than using y, mathematicians use the notation f(x) to denote the relationship between the inputs and outputs.

x represents the inputs, or independent variables

f(x) represents the outputs, or dependent variables

Domain

The domain of the function refers to the set of values the independent variable of a function can have. If you think of a function as a machine that spits out numbers, the domain specifies the numbers that can be entered into the machine. In the illustration on the left, the domain is made up of spheres.

Range

The range consists of the set of values of the dependent variable corresponding to the domain. If you think of a function as a machine that spits out numbers, the range is all the numbers that the machine can produce. In the illustration on the left, the machine took the inputs, or spheres, and produced the outputs as squares.

Unit 1-2: Types of Functions

Basic definitions

These are some key definitions to remember from Algebra II. These continue to play a large role in all types of functions.

Relation - A relation is any set of ordered pairs.
y-intercept - The y-intercept is the value of a function when x = 0. It gives the point where the graph crosses the y-axis. A function can only have one y-intercept.
x-intercept - The x-intercept is the value of x when y = 0. Keep in mind that functions can have more than one x-intercept.

Types of Functions

You learned all of these functions over the course of Algebra II last year. You can use these examples to remember what every function looks like, and you can see the corresponding equation for each function here. It is important that you have a strong understanding of these functions and equations so you can comprehend them on a more conceptual level during your time in Pre-Calculus*.

Polynomial Function

f(x) = anx^n + an-1xn-1 +... + a1x + a0, where n is a nonnegative number

Quadratic Function

f(x) = ax^2+bx+c, a ≠ 0

Linear Function

f(x) = ax+b

Direct Variation Function

f(x) = ax

Power Function

f(x) = ax^b

Exponential Function

f(x) = a * b^x

Inverse Variation Function

f(x) = a/x

Rational Algebraic Function

f(x) = n(x) / d(x)

Unit 1-3: Transformations

Dilations

The graph of a function can be dilated (magnified) by a certain factor in either the vertical or horizontal direction.

To vertically dilate a function, multiply the outputs, also known as the y-coordinates, by the factor. This is known as an outside transformation because it is done to the outside of the parentheses, or the value of the function. Vertical dilations have a narrowing effect on the shape of the graph. (see left)

For example, to magnify a function vertically by 3...

if f(x) = x², then... y = 3 f(x)

To horizontally dilate a function, multiply the x-coordinates, or inputs, by the reciprocal of the factor. This is known as an inside transformation because it is done to the inside of the parentheses, or the argument. Horizontal dilations stretch the graph side-to-side, widening the overall shape.

For example, to magnify the same function above horizontally by 3...

if f(x) = x^2, then... y = f(1/3 x)

Translations

The graph of a function can also be translated, or shifted, horizontally and vertically.

To vertically translate a function, simply add the intended shift in units to the value of the function:

y = f(x) + 3

To horizontally translate a function, subtract the intended shift in units to the argument, inside the parentheses, of the function:

y = f(x - 3)

You may have noticed that transformations affecting the x values seem to be counterintuitive. If you wanted to dilate the graph by a factor of 3, why would you multiply x by 1/3? To answer this question, you must think algebraically.

If you dilated a function horizontally by a factor of 3, each value of the argument would need to be three times what it was before the graph was magnified in order to generate the same y-values. Therefore:

y = f(g)

x = 3g

1/3x = g

Since g is equal to the argument of the function, you can substitute in 1/3x...

y = f(1/3x)

Unit 1-4: Composite Functions

You may remember composite functions from Algebra II, but this section will reinforce the information you learned last year so you can understand it on a deeper level during your time in Pre-Calculus*.

What is a Composite Function?

A composite function is a function that depends on another function. One function is substituted into another one.

Composite Function Example

A ripple of water has a radius that is increasing at 8 inches per second. What is the area of the radius after 5 seconds?

Upon reading the question, you would discover that area is a function of time.

Area depends on radius
Radius depends on time
- Since there are functions depending on other functions, the area of the ripple is a composite function.

Area can be written as a=(pi)(r)^2

The radius of the ripple is increasing at 8 in/s.
- Radius can be written as r=8t
At t=5, the radius is 40,
- With the radius at the questioned time, the area can be found using the area function.
  - The area is 5027in^2

Think about the numbers in correspondence to each function

5 is the input and 40 is the output for the radius function
40 is the input and 5027 is the output of the area function

These functions can be written as:

r(x)=8(x)
a(x)=(pi)(x)^2
- “r” and “a” become the names of the functions and r(x) and a(x) become the symbols
- “x” simply stands for the input of the function

By combining the symbols, you can make a composite function

Area=a(r(x))
- Function "r" is the inside function and function "a" is the outside function
  - The area of the ripple depends on the radius which depends on the time.

Composite Functions from Graphs

Composite functions from graphs are actually relatively simple. Take these two graphs, for example. The first graph, g(x), and the second graph, f(x), will be used for an example of composite functions from graphs.

Let's find f(g(30)) for our example.

First, you must find the value of the inside function.

g(30) is the inside function.
To find the answer to this function, you merely have to use the graph.
- Locate the x-axis on the g(x) graph, and find 30.
  - Now, find the point on the graph that uses 30 as the x-value.
    - To do this, simply find the y-value by using your eyes.
    - g(30) = 2.8

Next, you must complete the outside function.

f(g(30)) becomes f(2.8) because g(30) is 2.8
- In order to solve the outside function, and the composite function as a whole, you must use the f(x) graph.
  - Locate the 2.8 on the x-axis of the f(x) graph, and find the point on the line where the 2.8 is used as the x-value
    - Now, find the y-value that corresponds to the 2.8 as the x-value.
    - The y-value is 180, so f(2.8) = 180.

f(g(30)) = 180.

Using the graphs to solve the inside function and then the outside function allows you to solve the composite function.

Composite Functions from Tables

Though relatively simple as equations or graphs, composite functions are also manageable from tables.

Use this table for an example of how you can find the values of a composite function from a table.

Take f(1) for example.
- Instinctively, you should look at the x value column.
  - Find the 1 in that column, and then look over at the f(x) column.
    - The number in the f(x) column that is in the same row as the 1 in the x column is the answer to f(1).
      - The answer is 3.
Now, find the answer to a composite function using the table. Find f(g(6))
- Similarly to the function above, locate the 6 in the x column.
  - Now move over to the g(x) column, and see what number is in the same row as the 6 in the x column.
    - That number is 4.
      - Since the inside function equals 4, the function can be rewritten as f(4).
- This is the same thing as what you did in the first example. Locate the 4 in the x column and see what number is in the same row in the f(x) column.
  - The number is 2.
  - f(4) = 2
f(g(6)) = 2
- Solving composite functions from tables is relatively simple, but you just need to make sure you are finding the numbers corresponding to the x values column.

Domain and Range of Composite Functions

Domain and range in composite functions can be confusing, so this example will refresh your mind and allow you to remember how to work with two functions that may have different domains and ranges.

Look at the two graphs on the left. The left graph is g(x) and the right is f(x).

The first thing you should notice is that the domain and range of both functions differs. In order to help you navigate and solve this composite function, it is useful to create a table.
- The table below the graph models the domain and range of the two functions to help solve the composite function.

Notice the areas on the table that say NONE.

What does this mean?
- This means that the graph of either the g(x) or f(x) function does not have a point at that x-value.
- The g(x) graph could have a point, but the f(x) graph may not have that point. This means the composite function would not work in that domain and range.
  - How do you solve the domain of the function?

An easy way to do this is to graph the domains of the functions.

By seeing the domain of g(x) over the domain of f(x), you can see the intersection of the two functions. This intersection is the domain of the composite function.
- See the graphs to the left to visualize the intersection.

Unit 1-5: Inverse Functions

What is an inverse function?

If the input and output of a function can be switched and the resulting relation is also a function, they are referred to as inverse functions. In other words, the domain and range are switched.

The inverse of the function f(x) is denoted as f ^-1(x)

Do not confuse this symbol with taking the reciprocal of the function's value!

If both inverses are functions, they are referred to as invertible.

If you were to take the composition of a function and its inverse, you should get the original input as your answer. This is because the inverse of a function essentially undoes the function.

f -¹( f (x)) = x, provided x is in the domain of f and f (x) is in the domain of f ¹
f (f -¹(x)) = x, provided x is in the domain of f -¹and f -¹(x) is in the domain of f

A function that is strictly increasing or decreasing is a one-to-one function.

Question: Why would a function need to be one-to-one in order to have an inverse that is also a function?

Numerically

In a table, the inputs of f(x) should be the ouptuts of f ^-1(x) and vice-versa. Inverse functions essentially undo each other.

Algebraically

To solve for the inverse of a function algebraically, first replace f(x) with y. Switch x and y, then isolate y.

Tip: To check your work, plug in numbers from the range of the original function and see if you get the corresponding value from the domain of the original function.

Graphically

The original function and its inverse are always reflections of each other across the line y = x (green above).

Unit 1-6: Special Transformations

Reflections

Earlier, you learned about horizontal and vertical dilations. Did you ever wonder how the graph would change if the factor was -1? The result of multiplying the domain or range by -1 is a reflection over the y-axis and x-axis, respectively.

To vertically reflect the graph over the x-axis, multiply the value of the entire function by -1:

y = -f(x)

To horizontally reflect a graph over the y-axis, multiply the argument of the function by -1:

y = f(-x)

When reflecting a graph over the y-axis, keep in mind that you need to multiply the limits of the domain by -1 as well.

Even Functions

The function f is an even function if and only if f(-x) = f(x) for all x in the domain

Odd Functions

The function f is an odd function if and only if f(-x) = -f(x) for all x in the domain

Absolute Value Transformations

You can also take the absolute value of a function through its argument or value. The graph of the absolute value parent function is a symmetrical v-shape.

{f(x)} takes the absolute value of f(x)

Retains the non-negative values of y and reflects the negative values vertically across the x-axis

f({x}) takes the absolute value of the argument

For positive values of x, {x} = x
For negative values of x, {x} = -x

We hope that this page was helpful for your understanding of functions. Best of luck in the upcoming year!

Page updated

Report abuse