Table of Contents

  1. Introduction
  2. Definition of a Function
  3. Basic Function Terminology
  4. Evaluation of Functions
  5. Linear Function Example
  6. Quadratic Function Example
  7. Practice

Introduction

Functions are fundamental to algebra and mathematics in general. They describe how one quantity depends on another.

Definition of a Function

A function is a relation between a set of inputs and a set of permissible outputs. For each input, there is exactly one output. Functions are often expressed as \( f(x) $\)where \( x $\)is the input and \( f(x) $\)is the output.

Basic Function Terminology

  • Domain: The set of all possible input values.
  • Range: The set of all possible output values.
  • f(x): The function notation, where \( x $\)is the input and \( f(x) $\)is the output.

Evaluation of Functions

To evaluate a function means to find the output for a given input.

Linear Function Example

  • A linear function can be represented as \( f(x) = mx + b \).

  • Example: \( f(x) = 2x + 3 \), where \( m = 2 $\)and \( b = 3 \).

    f <- function(x) { return(2 * x + 3) }

    x_values <- seq(-10, 10, by = 0.1) y_values <- sapply(x_values, f)

    plot(x_values, y_values, type = "l", main = "Linear Function: f(x) = 2x + 3", xlab = "x", ylab = "f(x)", col = "blue")

Quadratic Function Example

  • A quadratic function is represented as \( f(x) = ax^2 + bx + c \).

This formula is a standard representation where \( a \), \( b \), and \( c $\)are constants, and aa is not zero. The function describes a parabola in the coordinate plane. The values of \( a \), \( b \), and \( c $\)determine the specific shape and position of the parabola.

  • Example: \( f(x) = x^2 - 4x + 3 \), with \( a = 1 \), \( b = -4 \), and \( c = 3 \).

    a <- 1 b <- -4 c <- 3

    f <- function(x) { return(a * x^2 + b * x + c) }

    x_values <- seq(-2, 5, by = 0.1) y_values <- sapply(x_values, f)

    plot(x_values, y_values, type = "l", main = "Quadratic Function: f(x) = x^2 - 4x + 3", xlab = "x", ylab = "f(x)", col = "red")

Practice

  1. Define a function \( g(x) = x^3 - 4x + 6 $\)and evaluate it at \( x = 2 \).
  2. For the function \( h(x) = 2x^2 - 5x + 3 \), find the output when \( x = -1 \).