Grade 9 · Mathematics
Quarter 4 · Week 5 · TEKS
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This Grade 9 mathematics lesson helps students decide whether relations represented verbally, tabularly, graphically, and symbolically define a function. Students will learn to analyze different representations of relations and determine if they meet the definition of a function. They will also practice evaluating functions using function notation given elements from their domains. This lesson aligns with TEKS standards for Algebra I and focuses on number and algebraic methods.
Students should understand the concept of relations as sets of ordered pairs and be familiar with coordinate planes, tables of values, and basic algebraic expressions.
A relation is a set of ordered pairs (x, y) where x is related to y. A function is a special type of relation where each input (x-value) corresponds to exactly one output (y-value). This means no x-value is paired with more than one y-value. Understanding this distinction is key to identifying functions.
Relations can be represented in multiple ways: verbally (described in words), tabularly (in tables), graphically (on coordinate planes), and symbolically (with equations or function notation). Each representation provides different clues to determine if the relation is a function.
When a relation is described in words, identify if each input has only one output. For example, "Each student has one unique ID number" describes a function because each student (input) has exactly one ID (output). Conversely, "A person can have multiple phone numbers" is not a function because one input (person) can have multiple outputs (phone numbers).
In a table of values, check the x-values (inputs). If any x-value repeats with different y-values, the relation is not a function. For example, if x=2 corresponds to y=3 and also y=5 in the table, it is not a function.
Use the vertical line test on the graph: if any vertical line intersects the graph more than once, the relation is not a function. This test checks if an x-value has more than one y-value. For example, a circle fails the vertical line test, so it is not a function.
When given an equation or function notation, analyze if for each x there is only one y. For example, y = x^2 is a function because each x gives one y. However, y^2 = x is not a function because for some x-values there are two y-values (positive and negative roots).
Function notation uses symbols like f(x) to represent functions. To evaluate f(a), substitute the input value a into the function expression and simplify to find the output. For example, if f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11.
Given the table: x: 1, 2, 3, 2; y: 4, 5, 6, 7. Is this relation a function?
Check if any x-value repeats with different y-values. Here, x=2 appears twice with y=5 and y=7, so one input has two outputs, which means it is not a function.
No, this relation is not a function because x=2 corresponds to two different y-values (5 and 7).
Use the vertical line test on the graph of a parabola opening upwards. Does it represent a function?
A parabola opening upwards passes the vertical line test because any vertical line intersects it at only one point, so each x has exactly one y.
Yes, the parabola represents a function because it passes the vertical line test.
Evaluate f(x) = 3x - 2 for x = 4.
Substitute 4 for x: f(4) = 3(4) - 2 = 12 - 2 = 10.
f(4) = 10.
Present a real-world scenario verbally describing a relation, such as assigning locker numbers to students. Ask students if this relation is a function and why.
Discuss with a partner whether the scenario describes a function and share reasoning with the class.
This activity connects students' prior knowledge of real-world relationships to the mathematical definition of functions, activating interest and context.
Provide tables of values representing different relations. Guide students to identify which tables represent functions by checking for repeated x-values with different y-values.
Work in pairs to analyze tables and decide if each is a function, explaining their reasoning.
This hands-on activity helps students practice identifying functions from tabular data, reinforcing the definition of a function.
Demonstrate the vertical line test on graphs of different relations. Show graphs that pass and fail the test.
Use graph paper or graphing tools to draw relations and apply the vertical line test to decide if they are functions.
The vertical line test is a visual method to determine if a graph represents a function by checking if any vertical line intersects the graph more than once.
Provide symbolic relations and function notation expressions. Guide students to determine if the relations are functions and evaluate the functions for given inputs.
Analyze symbolic relations to decide if they define functions. Evaluate function notation expressions for specified domain values.
This phase deepens understanding by connecting symbolic representations to the function concept and practicing function evaluation.
Distribute an exit ticket with mixed representations of relations. Ask students to decide if each is a function and evaluate given function notation problems.
Complete the exit ticket independently, demonstrating understanding of identifying functions and evaluating functions.
The exit ticket assesses students' mastery of deciding if relations are functions and evaluating functions, providing feedback for instruction.
Work in pairs to analyze tables and decide if each is a function, explaining their reasoning.
What is a function?
A function is a relation where each input has exactly one output.
Given the table: x: 1, 2, 3; y: 4, 5, 6. Is this a function? Explain.
Yes, because each x-value corresponds to exactly one y-value.
Use the vertical line test to determine if the graph of a circle is a function.
The circle fails the vertical line test because some vertical lines intersect it more than once, so it is not a function.
Given the equation y^2 = x, explain why it does not define y as a function of x.
For some x-values, there are two y-values (positive and negative roots), so one input corresponds to multiple outputs, meaning it is not a function.
Evaluate the function f(x) = 2x^2 - 3x + 1 for x = -2 and explain each step.
Substitute -2 for x: f(-2) = 2(-2)^2 - 3(-2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15.
Provide step-by-step examples with visual aids. Use manipulatives to represent ordered pairs. Offer guided practice with immediate feedback.
Use clear, simple language and visuals to explain functions. Provide vocabulary cards with definitions and examples. Encourage peer discussion to reinforce concepts.
Challenge students to write their own relations and test if they are functions. Explore inverse functions and their properties. Analyze more complex function notation and piecewise functions.
Ask students to find examples of functions and non-functions in their daily lives, such as schedules, recipes, or phone directories, and explain why they are or are not functions.
In this lesson, students learned to decide whether relations represented verbally, tabularly, graphically, and symbolically define a function. They practiced using the vertical line test on graphs, checking tables for repeated inputs, analyzing verbal descriptions, and evaluating functions using function notation. Understanding these concepts helps students recognize functions in various forms and apply function evaluation skills.
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