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.