Unlocking Functions: Input, Output, and the Magic in Between

Algebra 1 Grades High School 7:57 Video
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Lesson Description

This lesson introduces the concept of a function as a rule that assigns a unique output to each input. We'll explore function notation, identify functions from relations, and see how functions are used in real-world scenarios.

Video Resource

What is a function? | Functions and their graphs | Algebra II | Khan Academy

Khan Academy

Duration: 7:57
Watch on YouTube

Key Concepts

  • Definition of a Function: A function assigns each input to exactly one output.
  • Function Notation: Understanding and using f(x) notation.
  • Relations vs. Functions: Distinguishing between relations and functions using the vertical line test and input-output relationships.

Learning Objectives

  • Students will be able to define a function and explain its key characteristics.
  • Students will be able to use function notation to evaluate functions for given inputs.
  • Students will be able to differentiate between relations and functions using various representations (equations, graphs).
  • Students will be able to identify when a relation is not a function.

Educator Instructions

  • Introduction (5 mins)
    Begin by asking students what they think a 'function' is. Write their ideas on the board. Explain that a function is like a machine: you put something in (input), and it gives you something back (output). The key is that for each input, you only get one specific output.
  • Video Viewing (7 mins)
    Play the Khan Academy video 'What is a function?' Students should take notes on the definition of a function, function notation, and examples provided.
  • Guided Practice (10 mins)
    Work through examples similar to those in the video on the board. Start with simple numerical functions (e.g., f(x) = x + 2) and progress to more complex examples (e.g., piecewise functions). Emphasize the importance of substituting the input value for the variable in the function.
  • Identifying Functions (10 mins)
    Present students with a variety of relations (equations and graphs). Guide them in determining whether each relation is a function. Introduce the vertical line test as a visual method for identifying functions from graphs. Discuss the circle example from the video, explaining why it is not a function.
  • Independent Practice (8 mins)
    Students work individually or in pairs on a worksheet with problems that require them to evaluate functions, identify functions from relations, and explain their reasoning.

Interactive Exercises

  • Function Machine Activity
    Create a physical 'function machine' (a box with an input slot and an output slot). Write different functions on cards (e.g., 'Add 3,' 'Multiply by 2'). Students input a number, follow the function rule on the card, and write the output on a separate card. This activity provides a hands-on way to understand the input-output relationship of a function.
  • Graphing Functions Practice
    Students use graphing calculators or online graphing tools to graph various functions. They then analyze the graphs to determine if they represent functions and identify key features like intercepts and slope.

Discussion Questions

  • Can you think of real-world examples of functions?
  • What is the difference between a relation and a function? Why is this difference important?
  • How can the vertical line test help you determine if a graph represents a function?

Skills Developed

  • Critical Thinking: Analyzing and evaluating relationships to determine if they are functions.
  • Problem-Solving: Applying function notation to solve mathematical problems.
  • Abstract Reasoning: Understanding the concept of a function as an abstract relationship between inputs and outputs.

Multiple Choice Questions

Question 1:

Which of the following is the best definition of a function?

Correct Answer: A rule that assigns each input to exactly one output

Question 2:

If f(x) = 2x - 1, what is f(3)?

Correct Answer: 5

Question 3:

Which of the following graphs represents a function?

Correct Answer: A parabola

Question 4:

What is the input of a function commonly referred to?

Correct Answer: X

Question 5:

What is the output of a function commonly referred to?

Correct Answer: Y

Question 6:

In the function f(x) = x + 5, what happens to the input?

Correct Answer: It is added to 5

Question 7:

A function has how many possible outputs for each input?

Correct Answer: One

Question 8:

Which of the following examples is NOT a function?

Correct Answer: x^2 + y^2 = 1

Question 9:

What is the range of a function?

Correct Answer: The set of all output values

Question 10:

In function notation, f(x), what does 'f' represent?

Correct Answer: The name of the function

Fill in the Blank Questions

Question 1:

A _________ is a rule that assigns each input to exactly one output.

Correct Answer: function

Question 2:

In the function f(x) = x^2, if x = 4, then f(x) = _________.

Correct Answer: 16

Question 3:

The set of all possible input values for a function is called the _________.

Correct Answer: domain

Question 4:

If a vertical line intersects a graph more than once, the graph does NOT represent a _________.

Correct Answer: function

Question 5:

f(x) is a way of writing a _________.

Correct Answer: function

Question 6:

The y-value of f(x) is also known as the _________.

Correct Answer: output

Question 7:

In f(x) = 5x - 3, the number 5 is a _________ of x.

Correct Answer: coefficient

Question 8:

When evaluating a function, replace the variable with the _________ value.

Correct Answer: input

Question 9:

A relation is not a _________ if the same input can have multiple outputs.

Correct Answer: function

Question 10:

A line that only crosses the graph of a function once is _________.

Correct Answer: linear

Educational Standards

A1.F.1:[Standard] Understand functions as descriptions of covariation (how related quantities vary together) in real-world and mathematical problems. A1.F.1.1:[Objective] Distinguish between relations and functions. A1.F.1.2:[Objective] Identify the dependent variable, independent variable, domain and range given a function, equation, or graph. Identify restrictions on the domain and range in mathematical models. A1.F.3.2:[Objective] Use function notation; evaluate a function, including nonlinear, at a given point in its domain algebraically and graphically. Interpret the results in terms of the original context.

Teaching Materials

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