Lesson 4: Functions

Functions

​What is a Function?

​In a function, which is a type of relation (two values with some rule of correspondence), each x value has exactly 1 y value. A single y value can have more than on x value, however.

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​The domain is the set of x values

​The range is the set of y values

​Four Ways to Represent Functions

​We can represent functions in four ways:

​1. Verbally, with a sentence.

​2. Numerically, with a table or list.

​3. Graphically, with points in a coordinate plane.

​4. Algebraically, with an equation in two variables.

​Testing for Functions

​There are different tests for different ways of communicating functions:

​1. In a sentence, or a table, consider whether any given x value gives multiple y values at a time.

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​2. In a graph, use the vertical line test. If you can draw a vertical line through two points on the relation, then it is not a function.

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​3. Solve the equation for y. If there are two possible solutions of y for one given x, then it is not a function.

Multiple Choice

Does the table show a function?

• Yes

• No

​Function Notation

​We can use function notation to represent a function. Function notation looks like this:

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​Where f(x) is the output, x is the input, and the entire thing is the equation.

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​If we plug in an input, we replace both x's in the equation:

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​The first x is replaced to keep track of our input and then we solve the right side to find the output (y-value).

​Function Notation

Sometimes, expressions will be substituted into a function:

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​Let's substitute x + 4 into the function above:

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​Then, we will simplify in order to find our output:

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​Piecewise-Defined Functions

​Piecewise-defined functions have different expressions depending on the input. For the piecewise-defined function in the image, if x is less than or equal to 1, we use the expression on top (2x +3). If it is greater than 1, then we use the bottom expression (-x +4). Be sure to read the inequalities correctly or you'll use the wrong expression!

Multiple Choice

Which expression do I use for the indicated input, f(-1)?

• 4 - 5x

• 3x

• 4x + 1

Multiple Choice

What is the output of f(-1)?

• 9

• 1

• -1

​Domain of a Function

​The domain of a function changes depending on the function. In the function to the right, we cannot include any number less than 7 because that leads to a negative under the radical (which creates an imaginary number, not a real number). We also need to watch out for functions which may return undefined outputs (as is the case in some functions including x in the denominator of a fraction).

​Interval Notation

Sometimes, the range and domain are written in interval notation.​ For instance, the domain is written in interval notation as .

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​We use parentheses when the circle on the graph would be open (or, with < and >) and we use brackets if the circle on the graph would be closed . We also put infinity or negative infinity if there isn't an endpoint on that side. If there is an endpoint, then we would have an inequality like which, when written in interval notation, looks like (4, 7).

Multiple Choice

What value can not be included in the domain of the following function?

$$f\left(x\right)=\frac{9}{x-8}$$  

• 8

• 16

• 9

• 0

​Difference Quotient

​A basic definition in calculus uses the following ratio:

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​This ratio is called the difference quotient. We find the difference quotient by plugging in x + h and x to the given function and then solving.