Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Functions and Graphs → Types of Functions ↓


Piecewise Functions


In mathematics, piecewise functions are an essential concept, especially in grade 11 math. Understanding piecewise functions will help students make connections between algebraic expressions, graph shapes, and real-world scenarios where different rules apply at different intervals. This explanation will delve deep into the definition, methodology, examples, and applications of piecewise functions.

What are piecewise functions?

A piece-wise function is a function that is defined by several sub-functions, each of which applies to a specific interval in the domain. Instead of having a single expression for the entire domain, a piece-wise function has different expressions for different parts of its domain. This means that the rule for calculating the output changes depending on the input value.

Piecewise functions are usually written in the following format:

f(x) = {
    { expression1, condition1 }
    { expression2, condition2 }
    { expression3, condition3 }
    ,
,

Let us analyse this:

  • expression1, expression2, expression3,...: These are the various sub-functions or expressions that make up the piecewise function.
  • condition1, condition2, condition3,...: Each expression corresponds to a condition or interval of the domain for which the expression is valid.

A piece-wise function checks the condition for each x-value to decide which expression to use for that particular x-value.

Visualization of tasks in chunks

Let's take an example to explain this concept visually:

Consider a piecewise function that is defined as:

    f(x) = {
        { x + 2, x < 0 }
        { 3x, 0 ≤ x < 2 }
        { 2x – 1, x ≥ 2 }
    ,

This function has three parts or sections:

  • If x < 0, f(x) = x + 2 is a linear function with slope of 1 and y-intercept at 2.
  • If 0 ≤ x < 2, f(x) = 3x. This is also a linear function, but it has slope of 3 and passes through the origin.
  • If x ≥ 2, f(x) = 2x - 1 again, a linear function with a slope of 2 and a y-intercept at -1.

The above graph shows the piecewise function with three line segments, each of which is colored differently for clarity:

  • The red line shows f(x) = x + 2 for x < 0.
  • The green line represents f(x) = 3x for 0 ≤ x < 2.
  • The blue line shows f(x) = 2x - 1 for x ≥ 2.

Solving tasks in pieces

When a piecewise function is given, solving it simply means determining the value of the function for a particular x-value. The steps to solve a piecewise function can be outlined as follows:

  1. Determine which position or interval the given x-value falls in.
  2. Use the corresponding expression to calculate the value of the function.

Example 1

Let's solve the piecewise function defined earlier for several values of x:

f(x) = x + 2 where x < 0, f(x) = 3x where 0 ≤ x < 2, f(x) = 2x - 1 where x ≥ 2.

  • For x = -3: Since -3 < 0, use f(x) = x + 2.
    •  f(-3) = -3 + 2 = -1
  • For x = 0: Since 0 ≤ x < 2, use f(x) = 3x.
    •  f(0) = 3(0) = 0
  • For x = 1.5: Since 0 ≤ x < 2, use f(x) = 3x.
    •  f(1.5) = 3(1.5) = 4.5
  • For x = 2: Since x ≥ 2, use f(x) = 2x - 1.
    •  f(2) = 2(2) - 1 = 3
  • For x = 3: Since x ≥ 2, use f(x) = 2x - 1.
    •  f(3) = 2(3) - 1 = 5

Applications of piecewise functions

Piecewise functions are widely used in real-world applications. Here are some of the major areas where they are applied:

  • Tax brackets: Tax rates often vary at specific income levels, making taxes a perfect example of piecemeal taxation. Different portions of income may be taxed at different rates.
  • Shipping costs: Some shipping companies charge different rates based on weight. For example, the first kilogram may cost a certain amount, and additional kilograms have different rates.
  • Utility pricing: Electricity and water bills often include piecemeal components, where different slabs of consumption are billed at different rates.
  • Physics: In physics, many problems can be simplified by dividing tasks into pieces. For example, the motion of an object that moves at different speeds at different intervals.

Example 2: Real-world tax problem

Suppose the simplified tax system is defined as:

    Tax(Income) = {
        { 0.1 * income, 0 ≤ income ≤ 10000 }
        { 1000 + 0.2 * (Income - 10000), 10000 < Income ≤ 20000 }
        { 3000 + 0.3 * (Income - 20000), Income > 20000 }
    ,

This function indicates that:

  • The tax on income up to $10,000 is 10% of income.
  • For income between $10,001 and $20,000, the tax is $1000 plus 20% of the amount over $10,000.
  • The tax on income over $20,000 is $3,000 plus 30% of the amount over $20,000.

Calculation of tax for different incomes

  • For income of $9,000: Since 0 ≤ income ≤ 10000, use 0.1 * income.
    •  Tax(9000) = 0.1 * 9000 = 900
  • For income of $15,000: Since 10000 < income ≤ 20000, use 1000 + 0.2 * (income - 10000).
    •  Tax(15000) = 1000 + 0.2 * (15000 - 10000) = 1000 + 0.2 * 5000 = 2000
  • For an income of $30,000: Since income > 20000, use 3000 + 0.3 * (income - 20000).
    •  Tax(30000) = 3000 + 0.3 * (30000 - 20000) = 3000 + 0.3 * 10000 = 6000

Creating and interpreting functions piecewise

When you are given a scenario or problem, you can create a piece-wise function as follows:

  1. Identifying different gaps or situations that require different sub-tasks.
  2. Derive the expression corresponding to each interval or situation.
  3. Ensuring that the conditions cover all possible values for the input variable, usually x.

Example 3: Designing a custom piecewise function

Suppose we need to design a function for a graduate who receives a scholarship based on his academic achievements, defined as follows:

Scholarship Function:

    Scholarship (Grade Percentage) = {
        {$500, Grade Percentage < 60 }
        {$1000 + 10 * (grade percentage - 60), 60 ≤ grade percentage < 80 }
        {$2500, grade percentage ≥ 80 }
    ,
  • If the student's grade percentage is less than 60, they will be given an amount of $500.
  • If the grade percentage is between 60 and 80, they will receive $1,000 plus $10 for every percentage point above 60.
  • If the student scores 80% or more marks then he/she is given an amount of $2500.

Calculation of scholarship amount

  • For a grade percentage of 55: Use $500.
    •  Scholarships(55) = $500
  • For a grade percentage of 75: Use $1000 + 10 * (gradePercentage - 60).
    •  Scholarship (75) = $1000 + 10 * (75 – 60) = $1000 + 150 = $1150
  • For 85 grade percentage: Use $2500.
    •  Scholarships(85) = $2500

By understanding piecewise operations, students learn to break down complex scenarios into manageable components, analyze data, and determine outputs based on input conditions. The ability to interpret and construct piecewise operations is becoming increasingly valuable in mathematics and real-world problem-solving. As students advance in their mathematics education, they can expect to encounter more complex tasks that use piecewise definitions to describe various phenomena.


Grade 11 → 2.1.6


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Piecewise Functions

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