chapter 1 - functions

Chapter 1: Functions

Introduction to Functions

  • Functions are used as mathematical models to solve real-life problems such as predicting pandemic spread or inflation rates.

  • The simplest type of function is a linear function, commonly used in life sciences and agriculture due to its simplicity.

    • Examples include:

      • Light intensity vs. photosynthesis rate.

      • Glucose uptake vs. insulin level.

      • Cholesterol levels vs. heart disease risk.

Learning Objectives

By the end of this chapter, you should be able to:

  • Find function values.

  • Graph functions and determine whether a graph represents a function.

  • Find the domain and range of a function.

  • Evaluate and solve for a function.

  • Solve application problems involving functions.

Definition of a Function

  • A function is a rule that defines a relationship between input and output, converting each unique input into a unique output:

    • Each input must have an output.

    • For every input value, there is only one corresponding output value.

    • One-to-one function: One input corresponds to one output.

    • Many-to-one function: Several inputs relate to the same output.

  • Example in the natural world: Relationship between height and age.

Examples of Functions in Everyday Life

  • Grocery bills as a function of the items purchased.

  • Cooking time as a function of the thickness of the steak.

  • Travel time as a function of distance.

  • Income tax as a function of income.

Function Notation

  • We can represent functions using notation:

    • For height as a function of age, we denote it as h = f(a).

      • Here, h represents height and a represents age.

  • Example: A function expressed as N = f(y) can denote the number of koalas at a sanctuary as a function of the year.

Evaluating Functions

  • To evaluate a function, substitute known input values to find output values:

    • Example from a table of values where Q = g(n) for given n values.

Graphical Representation of Functions

  • Graphs can define functions with input values on the horizontal axis and output values on the vertical axis.

  • The vertical line test is used to determine if a graph is a function:

    • If a vertical line crosses the graph more than once, it is not a function.

Types of Functions

  1. Constant Function: f(x) = c

  2. Identity Function: f(x) = x

  3. Absolute Value Function: f(x) = |x|

  4. Quadratic Function: f(x) = x²

  5. Cubic Function: f(x) = x³

  6. Reciprocal Function: f(x) = 1/x

  7. Square Root Function: f(x) = √x

  8. Cube Root Function: f(x) = 3√x

Domain and Range

  • Domain: Set of possible input values for the function.

  • Range: Set of possible output values based on the domain.

  • Example: For a tree's height modeled as a function of circumference:

    • Domain: c > 0; maximum circumference of 40m.

    • Range: 0 < h ≤ 115m.

  • Limitations arise when dividing by zero or taking square roots of negative numbers.

Piece-Wise Functions

  • Defined by different formulae in different parts of the domain (e.g., parking fees per time).

  • Example of domain range for the piece-wise function is (0, 24].

Composition of Functions

  • Composition means substituting one function into another (e.g., f(g(x))).

  • The notation for composite function is h(x) = f(g(x)).

  • Restrictions from the inner and outer functions affect the domain and range of the composite function.

Practical Example

  • Metabolic Rate Function: Given by y = f(x) = 19.7x for anteaters where x is weight in kg.

  • Tasks:

    • Evaluate for specific weights.

    • Convert weight measurements (e.g., from pounds to kilograms).

    • Write metabolic rate as a function of weight in pounds.

Summary

  • Understanding functions is crucial in modeling relationships in real-world scenarios and evaluating their properties through various methods such as graphical representations and algebraic expressions.