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


In math, functions are like machines that take an input and give an output based on a specific rule. An inverse function is a function that exactly "undoes" the action of the original function. If the original function is represented as f(x), the inverse function is often represented as f -1 (x). Understanding inverse functions is an important part of math because they appear in a variety of scenarios such as solving equations, creating graphs, and real-world problems.

The concept of the inverse function

To understand the concept of an inverse function, consider the function f(x) = x + 2 Here, the rule is to add 2 to the input number. If f(x) = y, then y = x + 2 The inverse function, represented as f -1 (x), should take y and return the original input x . For this function, the inverse will be found by reversing the operation, leading to f -1 (x) = x - 2.

Let the function f(x) = x + 2 be
The inverse function will be f -1 (x) = x - 2

To confirm that they are inverses, we check if applying one function to the other returns us to the starting value. For example, if f(x) = 5, then x + 2 = 5 implies that x = 3. Using the inverse function, f -1 (5) = 5 - 2 = 3. You can see how the output is the input of the original function.

Graphical representation

Graphically, there is a special relationship between functions and their inverses. The graph of an inverse function is the reflection of the original function across the line y = x. Consider the function f(x) = 2x + 3 Its graph is a straight line with slope 2 that cuts the y-axis at 3. The inverse function f -1 (x) = (x - 3)/2 will reflect this line at y = x.

Original function: f(x) = 2x + 3
Inverse function: f -1 (x) = (x - 3)/2
Hey f(x) = 2x + 3 f - 1 (x) = (x - 3)/2

The blue line represents the function f(x) = 2x + 3, while the red line represents its inverse f -1 (x) = (x - 3)/2. The gray line is the reflection line y = x.

Finding the inverse function

To find the inverse function manually, follow these steps:

  1. Write the original function equation y = f(x).
  2. Switch the roles of x and y. This means rewriting the equation as x = f(y).
  3. Solve this new equation for y. The expression you get will be the inverse function.

Consider finding the inverse of f(x) = (3x - 4)/5 :

1. Start with y = (3x - 4)/5
2. Substitute x and y: x = (3y - 4)/5
3. Solve for y: 
   Multiply both sides by 5: 5x = 3y - 4
   Add 4 to both sides: 5x + 4 = 3y
   Divide by 3: y = (5x + 4)/3
Therefore, the inverse function is f -1 (x) = (5x + 4)/3

Properties of inverse function

The inverse function has several important properties:

  • Reflexive Property: The combination of a function and its inverse gives the identity function. For a function f, f(f -1 (x)) = x and f -1 (f(x)) = x.
  • Graphical symmetry: As stated earlier, the graph of a function and its inverse are symmetric about the line y = x.
  • Domain and range swap: The domain of the function f(x) becomes the range of f -1 (x), and vice versa.

Let us further look at how these properties manifest in real-life scenarios and applications.

Applications of inverse functions

Inverse functions are not just abstract concepts; they also have practical applications:

  • Solving equations: When an equation involves a function, finding its inverse can help isolate the variables and solve the equation. For example, if you need to solve 2^x = 16, using the inverse function of the exponent, which is the logarithm, can help find x.
  • Trigonometry: In trigonometry, inverse functions such as sin -1, cos -1, and tan -1 can be used to find the measure of an angle when a ratio is given.
  • Real-world problems: Inverse functions help convert units and reverse processes, such as converting Celsius to Fahrenheit and vice versa.

Consider a situation where you need to convert a temperature from Celsius to Fahrenheit or vice versa:

Celsius to Fahrenheit: f(x) = (9/5)x + 32
Fahrenheit to Celsius: f -1 (x) = (5/9)(x - 32)

These functions and their inverses are important in fields that involve heat and temperature conversion, such as meteorology and cooking.

More examples

Let us look at more examples to make our understanding of determining and verifying inverse functions even stronger.

Example 1: Quadratic function

Consider f(x) = x^2. At first glance, this function seems straightforward, but it has no inverse function that is also a function because it does not pass the horizontal line test, meaning that multiple y are possible for a single x.

Restrict the domain of f(x) = x^2 to non-negative numbers, make it a one-to-one function, then find the inverse:

y = x^2, where x >= 0
Swap x and y: x = y^2
Solve for y: y = √x

Thus, the inverse of f(x) = x^2 is f -1 (x) = √x for x >= 0.

Example 2: Exponential function

Consider an exponential function f(x) = e^x, where e is the base of the natural logarithm.

y = e^x
Swap x and y: x = e^y
Take the natural logarithm of both sides: ln(x) = y

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

Conclusion

Inverse functions play an important and pervasive role in mathematics, as they allow us to reverse processes and solve equations. By understanding the principles behind finding inverse functions and seeing how they correspond graphically, we not only build a solid foundation in algebra and calculus, but also open ourselves up to more advanced topics such as differential equations, complex analysis, and even practical fields such as engineering and physics.

By mastering inverse functions, you will deepen your problem-solving skills and enrich your mathematical understanding, which is essential for both academic pursuits and real-world applications.


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Inverse Functions

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