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 → Inverse Functions ↓


Finding Inverses


In mathematics, understanding the concept of an inverse function is crucial to understanding how functions work inversely. This topic delves deep into finding the inverse of a function, a fundamental concept that allows us to "undo" a given function. Through this detailed guide, we will explore what it means for a function to have an inverse, how to find this inverse, and numerous examples to solidify your understanding.

What is an inverse function?

The inverse function reverses the operation of the original function. If you have a function f that maps input x to output y, the inverse function, denoted as f -1, maps y back to x. The relationship can be expressed as:

f(x) = y

f -1 (y) = x

For an inverse function to exist, the original function must be bijective. This means it must be both one-to-one (injective) and onto (surjective).

How to determine if a function is bijective or not?

There are two main criteria for the inverse of a function:

  1. One-to-one (injective): Each output is mapped by only one input.
  2. Onto (surjective): Every possible output is covered by the function.

To determine whether a function is bijective, you can use these tests:

  • Horizontal Line Test: For the graph of a function, if each horizontal line intersects the graph at most once, then the function is one-to-one.

Example:

In the illustration above, the curve represents a function. The red line, which is horizontal, intersects the graph only once, indicating that the function is one-to-one. Therefore, it is a candidate for being the inverse.

Steps to find the inverse of a function

Follow these steps to find the inverse of a function:

  1. Change the function notation: Start with the function equation and replace f(x) with y.
  2. Swap variables: Interchange the variables x and y.
  3. Solve for y: Make sure all terms are isolated to solve for y.
  4. Replace y with f -1 (x): The new equation represents the inverse function.
  5. Verify: Verify that f(f -1 (x)) = x and f -1 (f(x)) = x.

Example 1: Linear function

Consider the function f(x) = 2x + 3.

  1. Replace f(x) with y:
     y = 2x + 3
  2. Replace x and y:
     x = 2y + 3
  3. Solve for y:
        x – 3 = 2y
    
    y = (x – 3) / 2
  4. Replace y with f -1 (x):
     f - 1 (x) = (x - 3) / 2

Thus, the inverse function is f -1 (x) = (x - 3) / 2.

Example 2: Quadratic function

Quadratic functions require careful consideration, since not all quadratic functions have inverses. Consider f(x) = x 2.

The function f(x) = x 2 is not one-to-one because f(2) = 4 and f(-2) = 4. To get the inverse, the domain can be restricted to make the function one-to-one (for example, x ≥ 0).

The above graph shows only the right half of the parabola, making f effectively one-to-one.

For this restricted function: f(x) = x 2, x ≥ 0

  1. Replace f(x) with y:
     y = x 2
  2. Replace x and y:
     x = y 2
  3. Solve for y:
     y = √x
  4. Replace y with f -1 (x):
     f -1 (x) = √x

Therefore, the inverse of the function is f -1 (x) = √x, for which x ≥ 0.

Relation between a function and its inverse

The inverse of a function is reflected on the graph at the line y = x. This reflection shows how the inputs and outputs are interchanged between the function and its inverse. Consider the example where f(x) = 2x + 3 and f -1 (x) = (x - 3) / 2. Graphically, they appear as reflections at the line y = x.

In the above graph, the gray dashed line is y = x, the blue line represents the function f(x), and the red line represents the inverse f -1 (x).

Additional examples

Example 3: Rational function

Suppose you have this function: f(x) = (2x + 3) / (x - 1).

  1. Replace f(x) with y:
     y = (2x + 3) / (x – 1)
  2. Replace x and y:
     x = (2y + 3) / (y – 1)
  3. Solve for y:
        x(y - 1) = 2y + 3
    
    xy−x=2y+3
    
    xy – 2y = x + 3
    
    y(x – 2) = x + 3
    
    y = (x + 3) / (x - 2)
  4. Replace y with f -1 (x):
     f - 1 (x) = (x + 3) / (x - 2)

Thus, the inverse is f -1 (x) = (x + 3) / (x - 2).

Example 4: Exponential function

For the exponential function f(x) = 2 x, the inverse is the logarithmic function.

  1. Replace f(x) with y:
     y = 2x
  2. Replace x and y:
     x = 2y
  3. Solve for y using logarithms:
        y = log 2 (x)
  4. Replace y with f -1 (x):
     f -1 (x) = log 2 (x)

Therefore, the inverse function is f -1 (x) = log 2 (x).

Practical application of inverse functions

Inverse functions have a wide range of applications in real-life scenarios, such as:

  • Science and Engineering: Solving equations involving temperature conversion, electricity, and fluid dynamics.
  • Cryptography: Many encryption algorithms rely on the concept of one-way functions and their inverses.
  • Economics: Inverse functions are often needed to calculate interest rates and growth models.

Conclusion

Understanding inverse functions is crucial in understanding the processes behind mathematical equations and real-world applications. They provide a strong tool for solving equations and understanding relationships between variables. Mastering inverse functions builds a foundation for more complex mathematical concepts and enhances critical thinking in problem-solving scenarios.


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Finding Inverses

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