Back to Algebra: What are Functions?

Summary

Professor Dave's lesson introduces the concept of functions in algebra, explaining how they relate input and output values. Using the example of a 30% discount, he illustrates how a function operates and emphasizes the importance of the vertical line test for function graphs. The lesson also covers domain and range, zeroes of functions, and how to evaluate a function with a specific value, providing a foundational understanding of algebraic functions.

Takeaways

• 📚 The typical math curriculum involves teaching a year of algebra, followed by a year of geometry, and then another year of algebra, often called algebra two.
• 🔍 A function relates two quantities: an input value and an output value.
• 📦 You can think of a function as a box that, when given an input value, produces an output value based on the function's rule.
• 🛒 For example, if a store has a 30% off sale, the function to find the sale price would be F(x) = 0.7x, where x is the original price.
• 📈 Functions can be represented in tables and graphs, showing the relationship between input and output values.
• 🧪 A key property of functions is that each input value must produce exactly one output value, which is tested using the vertical line test on graphs.
• 💡 Multiple input values can produce the same output value in a function, but the reverse is not allowed.
• 📐 The domain of a function is the set of all possible input values, while the range is the set of all possible output values.
• 🧮 Identifying zeroes of a function involves finding the input values that make the function equal to zero, which are the x-intercepts on a graph.
• 📝 Evaluating a function involves substituting a given input value into the function's formula to calculate the corresponding output value.

Q & A

What is the standard sequence of math subjects taught in schools according to the transcript?

The standard sequence is to teach a year of algebra first, followed by a year of geometry, and then another year of algebra, often referred to as algebra two.

What is the fundamental concept of a function in mathematics?

A function is a mathematical concept that relates two quantities: an input value and an output value. It can be visualized as a box that takes an input and produces an output based on the function's definition.

How does the transcript illustrate the concept of a function with a real-world example?

The transcript uses the example of a store offering a 30% discount on all items. The function to calculate the sale price is F(X) = 0.7X, where X is the original price and F(X) is the sale price.

What is the significance of the vertical line test in the context of functions?

The vertical line test is used to determine if a graph represents a function. A graph passes the test if any vertical line drawn through it intersects the graph at only one point, indicating a single output for each input.

Why is there no horizontal line test mentioned in the transcript for functions?

There is no horizontal line test because functions can have the same output for multiple inputs, meaning different X values can produce the same Y value.

What is the domain of a function?

The domain of a function is the set of all possible input values that can be used in the function. For example, in the function F(X) = 2X + 1, the domain is all real numbers.

What is the range of a function, and how does it differ from the domain?

The range of a function is the set of all possible output values that the function can produce. Unlike the domain, which often includes all real numbers, the range can vary significantly depending on the function and may not include all real numbers.

Why can't the value of X be zero in the function F(X) = 1/(X - 2)?

In the function F(X) = 1/(X - 2), X cannot be zero because it would result in division by zero, which is undefined in mathematics.

What are the zeroes of a function, and how can they be identified graphically?

The zeroes of a function are the input values that make the function equal to zero. Graphically, they can be identified as the X-intercepts where the graph of the function crosses the X-axis.

How is the function F(X) = 3X^2 - 5X + 2 evaluated for F(2) according to the transcript?

To evaluate F(2), replace X with 2 in the function: F(2) = 3(2)^2 - 5(2) + 2. This simplifies to F(2) = 3(4) - 10 + 2, which equals 12 - 10 + 2, resulting in F(2) = 4.

Outlines

00:00

📚 Introduction to Functions

Professor Dave introduces the concept of functions in mathematics, explaining them as a relationship between two quantities, an input and an output. He uses the example of a store discount to illustrate a function, where the original price is the input and the sale price is the output. The function F(x) = 0.7x is given as an example, and it's explained how to evaluate it by plugging in different values for x. The concept of the vertical line test is introduced to determine if a graph represents a function, emphasizing that a function must have only one output for each input. The explanation also covers the domain and range of functions, with examples to illustrate these concepts.

05:01

🔍 Identifying Function Zeroes and Evaluation

This paragraph delves into identifying the zeroes of a function, which are the input values that result in an output of zero, graphically represented as the X-intercepts. The paragraph also explains how to evaluate a function with a specific example, showing the process of substituting a value into the function to find the corresponding output. The importance of understanding function evaluation is highlighted, and the paragraph concludes with a prompt to check comprehension on the topic of functions.

Keywords

💡Function

A function in mathematics is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. It is central to the video's theme as it is used to explain the concept of mapping inputs to outputs, such as calculating the sale price of an item with a discount: 'F of X equals 0.7 X'.

💡Input Value

An input value is the value that is put into a function to receive an output. In the context of the video, input values are used to demonstrate how a function operates, such as inserting a price to find the discounted price: 'If we plug in one dollar for X, the function spits out seventy cents'.

💡Output Value

The output value is the result that a function produces when an input value is applied. It is essential to the video's explanation of functions, as it shows the outcome of the process, like the sale price after applying the discount function: 'F of one equals 0.7'.

💡Algebra

Algebra is a branch of mathematics concerning the study of values that can vary and the rules for manipulating these values. The video mentions the teaching sequence of algebra in schools, indicating its foundational role in understanding mathematical concepts, including functions.

💡Geometric Figures

Geometric figures are shapes in geometry, and the video mentions having done work with them, indicating a transition from geometry to algebra. This sets the stage for discussing functions, which can relate to both algebraic and geometric concepts.

💡Vertical Line Test

The vertical line test is a graphical method to determine if a curve is a function. If any vertical line intersects the curve at more than one point, it is not a function. The video uses this concept to explain the uniqueness of outputs for a given input in functions.

💡Domain

The domain of a function is the set of all possible input values. The video explains it in the context of permissible values that can be used in a function, such as all real numbers for 'F of X equals two X plus one', but not including two for 'F of X equals one over X minus two' due to division by zero.

💡Range

The range of a function is the set of all possible output values. The video illustrates this with examples, such as all real numbers for a linear function and non-negative values for the absolute value function, emphasizing the variability of range based on the function type.

💡Zeroes of a Function

The zeroes of a function are the input values that result in an output of zero. The video describes how to identify these graphically as the X-intercepts, where the function crosses the X-axis, providing a practical method to find such values.

💡Evaluate

To evaluate a function means to calculate the output for a given input. The video demonstrates this with the function 'three X squared minus five X plus two' by substituting 'two' for 'X' and performing the calculation, resulting in 'F of two equals four'.

💡Graph

A graph is a visual representation of the relationship between inputs and outputs in a function. The video uses graphs to illustrate the concept of functions, such as showing a line as a function and a circle as not a function due to the vertical line test.

Highlights

Introduction to the concept of a function in algebra.

Explanation of the standard math teaching sequence in schools.

Transition from geometry back to algebra to expand understanding.

Definition of a function as a relationship between an input and output value.

Illustration of a function using a 30% off sale example.

Introduction of the function notation F(x) = 0.7x.

Explanation of how to use the function to find the sale price.

Demonstration of creating a table for function values.

Introduction of the concept of graphing functions.

Explanation of the vertical line test for identifying functions.

Clarification that a function must have a unique output for each input.

Introduction of the concept of domain and range of a function.

Example of determining the domain of a simple function.

Explanation of limitations on the domain, such as division by zero.

Introduction of the concept of range and its determination.

Example of determining the range for a function with absolute value.

Identification of the zeroes of a function and their graphical representation.

Demonstration of evaluating a function for a specific input value.

Comprehensive check to assess understanding of functions.

Transcripts

It’s Professor Dave; let’s learn about functions.

The standard practice of teaching math in school is to teach a year of algebra first,

then a year of geometry, and then do another year of algebra, which is often called algebra two.

Since we’ve just done a lot of work with geometric figures, we are ready to come back

to algebra and expand our understanding of this subject.

The first thing we want to do is comprehend the idea of a function.

A function is something that relates two quantities, an input value, and an output value.

So we can think of it as a little box, and when we insert input values, it spits out

the output values, in a way that depends on what the function is.

Let’s say at a particular store, everything is thirty percent off today.

That means to find the new price of an item we would multiply the original cost by 0.7,

because that will give us seventy percent of the original price, which is the same thing

as thirty percent off, since one hundred minus thirty is seventy.

The function that we would therefore use to compute the sale price of any item would be

F of X equals 0.7 X. F is the function, and, the X in parentheses, which is read “of

X”, means that the function will operate on any X value that you plug in.

If we plug in one dollar for X, the function spits out seventy cents, so F of one equals 0.7.

F of two equals 1.4, F of three equals 2.1, and so forth.

We could make a table like this, and we could even graph the resulting relationship.

So everything we used to do with equations where Y depends on X, we can do the same thing

with functions.

The only difference is that with functions, for any X value we plug in, there must only

be one Y value.

For example, if a function tells us the sale price of an item, if we plug in the regular

price, we should get only one sale price.

For that reason, for a graph to be representative of a function, it must pass the vertical line test.

Since a function can only have one output, or Y value, for each unique input, or X value,

then if we move a vertical line across a function, it will only intersect that function at one

point, wherever the vertical line may be.

If the vertical line intersects at two or more spots, it is not a function.

That would be as though we plugged in the regular price of an item and got back two

different numbers for the sale price.

We can, however, have the same output for multiple inputs, which is why different X

values are permitted to produce the same value for the function, so there is no horizontal

line test necessary.

This line is a function.

This circle is not.

This curve is a function.

This curve is not.

We can also describe the domain and range of a function.

The domain is essentially all of the values that can be plugged into the function.

So if we have something simple like F of X equals two X plus one, the domain would be

all real numbers, or negative infinity to positive infinity.

All the function does to the input is double it and add one, so any number will work.

But there are examples where there are limitations to the domain.

Let’s say F of X equals one over X minus two.

We can’t divide something by zero, so X can’t be two.

If X was two, we would get zero on the bottom, and the function would be undefined, so two

is not part of the domain.

The range, on the other hand, is all the potential output values, or essentially the values that

the function can possess.

Again, something simple like F of X equals two X plus one will yield a range that includes

all real numbers, extending to infinity in each direction.

But if we have something like F of X equals the absolute value of X, then the range is

now limited to values greater than or equal to zero.

So while the domain is very frequently all real numbers, the range has a tendency to

vary quite a bit depending on the type of function we are looking at.

We will also sometimes want to identify the zeroes of a function.

These are the input values that make the function equal to zero, which we can identify graphically

as the X-intercepts.

Wherever the function crosses the X-axis, that value of X is producing a Y value of

zero, so that is a zero of the function.

Again, different types of functions will have different numbers of zeroes.

Before we look at different types of functions, we need to make sure we know how to evaluate

a function.

For the function three X squared minus five X plus two, what is F of two?

Well let’s rewrite this but with two in the parentheses instead of X.

That means we are evaluating F of two.

We then also put a two instead of X wherever there is an X.

Two squared is four, times three is twelve, five times two is ten, so twelve minus ten

plus two equals four.

F of two equals four.

Now that we know what functions are, let’s check comprehension.