# Back to Algebra: What are Functions?

TLDRThe script explains what a function is in math - a relationship between an input and an output value. Functions take an input, do something to it, and output a result. Examples are given, like a 30% off sale function. Graphs of functions are discussed, including how to tell if a graph represents a function using the vertical line test. The domain and range of functions are explained - domain is the set of allowable inputs, and range is the set of possible outputs. Identifying zeros of functions is discussed, as well as evaluating functions by substituting input values. The goal is to build understanding of this key concept before looking at different types of functions.

###### Takeaways

- π A function relates an input value to an output value, like a box that spits out output values depending on the input.
- π We can represent functions visually with tables and graphs, which must pass the vertical line test to be valid.
- π€ The domain of a function is all possible input values that can be plugged into the function.
- π§ The range is all possible output values the function can produce for the given inputs.
- π Zeroes of a function are input values that make the function equal zero.
- π€ To evaluate a function means to plug in a value for the input and compute the resulting output.
- π Types of functions include linear, quadratic, polynomial, rational, exponential, logarithmic, and trigonometric.
- β° Sequences and series involve listing output values of functions in order.
- π’ The algebra of functions involves combining, manipulating and transforming functions.
- π Understanding functions underpins much of higher math and its applications.

###### Q & A

### What is a function in math?

-A function is something that relates two quantities - an input value and an output value. It's like a box where when you insert input values, it spits out output values that depend on what the function is.

### What is the difference between a function and an equation?

-An equation shows a relationship between variables, while a function relates input values to output values. For a relation to be considered a function, each input value can only have one corresponding output value.

### What is the domain of a function?

-The domain of a function refers to all the possible input values that can be plugged into the function. For simple functions, the domain is usually all real numbers.

### What is the range of a function?

-The range of a function refers to all the potential output values the function can produce. The range depends on the form of the function and is not always all real numbers.

### How do you find the zeros of a function?

-The zeros of a function are the input values that make the function equal to zero. Graphically, the zeros are the x-intercepts, where the function crosses the x-axis.

### What is the vertical line test?

-The vertical line test says that for a graph to represent a function, any vertical line drawn on the graph can only intersect the curve once. If it intersects more than once, it is not a function.

### How do you evaluate a function?

-To evaluate a function, you plug in the input value wherever you see the variable, then simplify the expression to find the output value. For example, to evaluate f(2) = 3x^2 - 5x + 2, plug in 2 for x to get f(2) = 3(2)^2 - 5(2) + 2 = 4.

### What are some examples of functions?

-Common examples of functions include linear functions like f(x) = 2x + 1, quadratic functions like f(x) = x^2, rational functions like f(x) = 1/x, and trigonometric functions like f(x) = sin(x).

### Can a function have the same output for different inputs?

-Yes, a function can produce the same output value for different inputs. For example, f(x) = |x| has f(-2) = f(2) = 2. The key is that each input has only one output.

### What is the difference between a function and not a function graphically?

-Graphically, a function passes the vertical line test - any vertical line intersect the curve at most once. A non-function fails this test and has vertical lines that intersect the curve more than once.

###### Outlines

##### π Introducing Functions in Algebra

This paragraph introduces the concept of a function in algebra. It explains that a function relates an input value to an output value, and provides examples of functions like calculating a 30% discounted price. It states that functions must produce only one output for each input, which is why functions must pass the vertical line test when graphed. It also distinguishes between the domain and range of a function.

##### π Evaluating Functions and Identifying Zeroes

This paragraph explains how to evaluate functions by substituting input values and calculating the output. It provides an example of evaluating F(2) for a specific function. It also introduces the idea of identifying zeroes of a function, which are input values that make the function equal zero. These are represented by x-intercepts on a graph. The paragraph states that different function types can have different numbers of zeroes.

###### Mindmap

###### Keywords

##### π‘function

##### π‘domain

##### π‘range

##### π‘vertical line test

##### π‘zeroes

##### π‘evaluate

##### π‘graph

##### π‘algebra

##### π‘undefined

##### π‘real numbers

###### Highlights

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###### Transcripts

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