Functions and Function Notation

Grade 8 · Math · Study Guide

Understand what makes a relation a function and how to use f(x) notation to evaluate and interpret functions.

A function is a relationship between two sets where each input has exactly one output. The input is called the domain and the output is called the range. You can think of a function like a machine: you put something in (input/x), and exactly one thing comes out (output/y).

A relation is a function if every x-value (input) is paired with exactly one y-value (output). The vertical line test is a quick visual check: if any vertical line drawn on a graph crosses the graph more than once, the graph does NOT represent a function.

Function notation uses f(x), read as 'f of x,' to describe the output when x is the input. For example, if f(x) = 2x + 3, then f(4) means substitute 4 for x: f(4) = 2(4) + 3 = 11.

Functions can be represented as equations, tables, graphs, or mappings. A mapping diagram shows arrows from each input to its output — a function has exactly one arrow leaving each input.

Linear functions have the form f(x) = mx + b and their graphs are straight lines. The rate of change (slope) of a linear function is constant.

Key terms

Function
A relation where each input has exactly one output
Domain
The set of all possible input values (x-values) of a function
Range
The set of all possible output values (y-values) of a function
f(x) notation
Function notation read 'f of x'; represents the output when x is the input
Vertical line test
A graph represents a function if no vertical line crosses it more than once
Mapping diagram
A diagram showing arrows from each input to its output; a function has one arrow per input

Flashcards

Function
Each input (x) has exactly one output (y)
Not a function
One input maps to two or more different outputs
Domain
All possible x-values (inputs)
Range
All possible y-values (outputs)
f(x) notation
Read 'f of x'; replace x with a number to evaluate the function
Evaluating a function
Substituting a value for x and simplifying; e.g., f(3) means plug in x = 3
Vertical line test
A graph is a function if no vertical line touches it more than once
Linear function
f(x) = mx + b; graph is a straight line with constant slope

Questions

  1. If f(x) = 3x - 5, what is f(4)?

    1. 7
    2. 12
    3. 17
    4. -2
    Show answer

    Answer: 7

    f(4) = 3(4) - 5 = 12 - 5 = 7.

  2. A relation where the input 3 maps to both 7 and 9 is a function.

    Show answer

    Answer: False

    A function requires each input to have exactly ONE output. If 3 maps to both 7 and 9, it violates the definition.

  3. The set of all input values of a function is called the _____.

    Show answer

    Answer: domain

    The domain is the set of all valid input values (x-values) for a function.

  4. Which statement best describes the vertical line test?

    1. If a vertical line crosses the graph more than once, it is a function
    2. If a vertical line crosses the graph more than once, it is not a function
    3. If a horizontal line crosses the graph more than once, it is not a function
    4. Vertical lines are always functions
    Show answer

    Answer: If a vertical line crosses the graph more than once, it is not a function

    If a vertical line hits the graph at two points, the same x-value has two y-values, violating the function definition.

  5. f(x) = 5x² is a function because each x-value produces exactly one output.

    Show answer

    Answer: True

    For any real number x, x² gives one specific value, so 5x² has exactly one output for each input.

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