### Overview of Covered Topics

In this lesson, the relationships between two variables, specifically functions, will be revisited. The lesson will address the following key points:

- Determining Whether a Relation Represents a Function
- Functions Defined by Sets of Ordered Pairs
- Functions Defined by Equations
- Finding Input and Output Values of a Function
- The Importance of Function Notation

#### Determining Whether a Relation Represents a Function

In mathematics, a function is a specific type of relationship where each input is associated with only one output. This concept is essential, as the output is determined by the input(s).

Examples of familiar functions include:

**Area of a Rectangle**: A=LWA = LWA=LW (Inputs: L=L =L= Length, W=W =W= Width; Output: A=A =A= Area)**Perimeter of a Square**: P=4SP = 4SP=4S (Input: S=S =S= Side Length; Output: P=P =P= Perimeter)**Volume of a Right Circular Cylinder**: V=πr2hV = \pi r^2 hV=πr2h (Inputs: r=r =r= Radius, h=h =h= Height; Output: V=V =V= Volume)

To further understand the concept of a function, consider a calculator: when the square root key is pressed, it returns the square root of the input (e.g., input 16, output 4). The calculator’s consistent output for each input exemplifies a function.

**Term to Know**: *Function* – A correspondence between a set of inputs, xxx, and a set of outputs, yyy, where each input corresponds to at most one output.

##### Functions Defined by Sets of Ordered Pairs

A function can be represented by a collection of ordered pairs (x,y)(x, y)(x,y), where xxx is the input and yyy is the output.

**Example**:

- Let xxx represent the temperature in Celsius and yyy the corresponding temperature in Fahrenheit. Examples of ordered pairs: (0,32)(0, 32)(0,32), (100,212)(100, 212)(100,212), (37,98.6)(37, 98.6)(37,98.6), (−40,−40)(-40, -40)(−40,−40). This relationship is a function since each Celsius temperature has a unique Fahrenheit equivalent.

Another example involving students and homework:

- Let xxx be the number of assignments completed and yyy the quiz score. Ordered pairs for several students might be {(4,15),(3,12),(5,18),(4,10),(3,16),(2,10),(5,14),(4,16)}\{(4, 15), (3, 12), (5, 18), (4, 10), (3, 16), (2, 10), (5, 14), (4, 16)\}{(4,15),(3,12),(5,18),(4,10),(3,16),(2,10),(5,14),(4,16)}. This relationship is not a function since some inputs (e.g., x=4x = 4x=4) correspond to multiple outputs (15, 10, 16).

##### Functions Defined by Equations

This course primarily deals with equations involving two variables. Generally, xxx is the input variable and yyy the output variable. When an equation defines a function, yyy is a function of xxx.

**Examples**:

- y=2x+3y = 2x + 3y=2x+3: This is a function as each input xxx yields a unique output yyy. Example ordered pairs: (0,3)(0, 3)(0,3), (1,5)(1, 5)(1,5), (2,7)(2, 7)(2,7), (3,9)(3, 9)(3,9).
- x2+y2=25x^2 + y^2 = 25×2+y2=25: This is not a function because some xxx-values correspond to multiple yyy-values, e.g., (0,5)(0, 5)(0,5) and (0,−5)(0, -5)(0,−5).
- y=x2−4x−2y = x^2 – 4x – 2y=x2−4x−2: This is a function as each xxx corresponds to a single yyy. Example ordered pairs: (1,−5)(1, -5)(1,−5), (2,−6)(2, -6)(2,−6), (3,−5)(3, -5)(3,−5), (4,−2)(4, -2)(4,−2).

**Big Idea**: Any equation that can be written without using the plus-or-minus sign (±\pm±) defines a function.

**Example**: For the equation 2x−3y=62x – 3y = 62x−3y=6, solving for yyy gives y=23x−2y = \frac{2}{3}x – 2y=32x−2. Since there is no plus-or-minus, yyy is a function of xxx.

#### Finding Input and Output Values of a Function

When a relation qualifies as a function, it can be expressed using function notation. For instance, y=x2y = x^2y=x2 can be written as f(x)=x2f(x) = x^2f(x)=x2. Here, xxx is the input variable, and f(x)f(x)f(x) is the output variable.

**Example**: For f(x)=x2f(x) = x^2f(x)=x2, find f(2)f(2)f(2), f(−4)f(-4)f(−4), and f(a+3)f(a+3)f(a+3):

- f(2)=22=4f(2) = 2^2 = 4f(2)=22=4
- f(−4)=(−4)2=16f(-4) = (-4)^2 = 16f(−4)=(−4)2=16
- f(a+3)=(a+3)2=a2+6a+9f(a+3) = (a+3)^2 = a^2 + 6a + 9f(a+3)=(a+3)2=a2+6a+9

**Hint**: Ensure the entire input is squared.

**Try It**: Use f(x)=x2−4x+2f(x) = x^2 – 4x + 2f(x)=x2−4x+2 to find f(1)f(1)f(1), f(−3)f(-3)f(−3), and f(a+1)f(a+1)f(a+1).

#### The Importance of Function Notation

Function notation is useful for computing different values from a single input.

**Example**: For a square with side length xxx:

- Area: A(x)=x2A(x) = x^2A(x)=x2
- Perimeter: P(x)=4xP(x) = 4xP(x)=4x
- Diagonal: D(x)=x2D(x) = x\sqrt{2}D(x)=x2

For a square of side 5 inches:

- Area: A(5)=25A(5) = 25A(5)=25 square inches
- Perimeter: P(5)=20P(5) = 20P(5)=20 inches
- Diagonal: D(5)=52D(5) = 5\sqrt{2}D(5)=52 inches

Function names (A, P, D) provide meaningful output names beyond yyy.

**Hint**: The input variable can vary, e.g., height of a projectile over time h(t)h(t)h(t).

### Summary

This lesson covered the determination of whether a relation is a function, emphasizing that each input must correspond to only one output. Functions can be defined by sets of ordered pairs (x,y)(x, y)(x,y) or equations where yyy is a function of xxx. Function notation highlights the relationship between input and output values, with xxx as the input variable and f(x)f(x)f(x) as the output variable.