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• A relation associates each element of set (A) with at least one element of set (B).
• An element in set (A) can map to multiple elements in set (B) in a relation.

Function

• A function is a relation where each element in set (A) (the domain) is associated with exactly one element in set (B) (the range).
• For every input, there is a unique output.

Types of Functions

• One-to-One Function: Each element of the domain maps to a unique element of the range.
• Many-to-One Function: Multiple elements of the domain map to the same element of the range.

Graphical Representation

• One-to-One Function: Every vertical line intersects the graph at most once.
• Many-to-One Function: Every vertical line intersects the graph at most once, but different vertical lines can intersect the graph at the same (y)-value.
• One-to-Many Relation (Not a Function): At least one vertical line intersects the graph more than once.

Definition

• An inverse function reverses the operation of a given function.
• For a function (f) with domain (X), the inverse function (f^{-1}) satisfies the following for every (x) in (X): [ f^{-1}(f(x)) = x ]
• (f) and (f^{-1}) undo each other's operations.
• Function: (y = f(x))
• Inverse: (x = f^{-1}(y))

Key Properties

• One-to-One Relation: A function must be one-to-one for its inverse to also be a function.
• If a function is not one-to-one, its inverse cannot be uniquely defined.
• Graphical Symmetry: The graph of the inverse function is the reflection of the original function's graph across the line (y = x).

Finding the Inverse Function

• Interchange (x) and (y) in the equation (y = f(x)).
• Solve for (y) to get the equation in terms of (y).

Graphical Representation

• When graphed, (f(x)) and (f^{-1}(x)) are symmetrical about the line (y = x).

Important Note

• The notation (f^{-1}(x)) indicates the inverse function and not the reciprocal of (f(x)).
• ((f(x))^{-1}) represents the reciprocal.

Horizontal Line Test

• One-to-One: Every horizontal line intersects the graph at most once.
• Many-to-One: A horizontal line intersects the graph more than once.

Inverse of Linear Functions

• The inverse of a linear function (f(x) = ax + q) reverses the effect of (f).
• It swaps the roles of the dependent and independent variables, interchanging the domain and range.

Steps to Find the Inverse of a Linear Function

• Start with the function: [ y = ax + q ]
• Interchange (x) and (y): [ x = ay + q ]
• Solve for (y): [ y = \frac{1}{a}x - \frac{q}{a} ]
• Express in function notation as (f^{-1}(x)): [ f^{-1}(x) = \frac{1}{a}x - \frac{q}{a} ]

General Form

• For the function (y = ax + q), the inverse is (y = \frac{1}{a}x - \frac{q}{a}).
• The inverse of a linear function is also a linear function.
• The domain and range of the inverse function are (\mathbb{R}), assuming (a \neq 0).

Inverse of the Function (y = ax^2)

• Interchange (x) and (y) in the equation.
• Solve for (y).
• Express the new equation in function notation.

General Form for Inverse of (y = ax^2)

• To find the inverse of a quadratic function (y = ax^2), follow these steps: [ x = a y^2 ] [ y = \pm \sqrt{\frac{x}{a}} ]
• For the inverse to be a function, restrict the domain: [ y = \sqrt{\frac{x}{a}} \quad \text{if } x \ge 0 ]

Properties of the Inverse

• Domain and Range: The domain of the original function becomes the range of the inverse function and vice versa.

• Restrictions:

  • If a quadratic function is not naturally one-to-one, restrict its domain so its inverse can be a function.
  • For (y = ax^2):
    • If (a > 0), typically restrict (x \geq 0).
    • If (a < 0), typically restrict (x \leq 0).

• (\text{For } f(x) = -x^2):

  • (\text{x-intercepts}: x = 0.)
  • (\text{y-intercepts}: y = 0.)

• (\text{For } f^{-1}(x) = \sqrt{-x}):

  • (\text{x-intercepts}: x = 0.)
  • (\text{y-intercepts}: y = 0.)

• Sketch both graphs and ensure they are reflections about the line (y = x).

Revision of Exponents

• An exponent indicates the number of times a base number is multiplied by itself.

Graphs of the Exponential Function (f(x) = b^x)

• (\mathbf{b > 1}): The function is increasing.

• (\mathbf{0 < b < 1}): The function is decreasing.

• (\mathbf{b \leq 0}): The function is not defined.

• (\mathbf{b > 1}): The graph rises rapidly.

• (\mathbf{0 < b < 1}): The graph falls rapidly.

• (\mathbf{b \leq 0}): The function is not applicable.

Inverse of the Function (y = b^x)

• (x = b^y)
• (y = \log_b(x))
• The inverse of (y = b^x) is (y = \log_b x).

Definition of Logarithm

• If (x = b^y), then (y = \log_b(x)).
• The logarithm of a number (x) with a base (b) is the exponent (y) to which (b) must be raised to yield (x).

Converting Between Exponential and Logarithmic Forms

• Exponential to Logarithmic: [ 5^2 = 25 \quad \longrightarrow \quad \log_5 25 = 2 ]
• Logarithmic to Exponential: [ \log_2 128 = 7 \quad \longrightarrow \quad 2^7 = 128 ]

Properties and Graphs of Exponential and Logarithmic Functions

• Exponential Function (y = b^x):
  • Shape: Increasing or decreasing.
  • Intercept: ((0, 1)).
  • Asymptote: Horizontal asymptote at (y = 0).
  • Domain: (x \in \mathbb{R}).
  • Range: (y > 0).
• Logarithmic Function (y = \log_b x):
  • Shape: Increasing.
  • Intercept: ((1, 0)).
  • Asymptote: Vertical asymptote at (x = 0).
  • Domain: (x > 0).
  • Range: (y \in \mathbb{R}).

Special Logarithmic Values

• (\log_a 1 = 0) because (a^0 = 1)
• (\log_a a = 1) because (a^1 = a)

Laws of Logarithms

• Product Rule: (\log_a(xy) = \log_a x + \log_a y) (For (x, y > 0))
• Quotient Rule: (\log_a \bigl(\frac{x}{y}\bigr) = \log_a x - \log_a y) (For (x, y > 0))
• Power Rule: (\log_a(x^b) = b \log_a x) (For (x > 0))
• Change of Base: (\log_a x = \frac{\log_b x}{\log_b a}) (For (b > 0))

Graphs and Inverses of Exponential and Logarithmic Functions

• The graph of the inverse is the reflection of the original function about the line (y = x).
• Exponential (f(x) = 10^x):
  • Intercept: ((0,1)).
  • Asymptote: (y = 0).
  • Domain: (x \in \mathbb{R}).
  • Range: (y > 0).
• Logarithmic (f^{-1}(x) = \log x):
  • Intercept: ((1,0)).
  • Asymptote: (x = 0).
  • Domain: (x > 0).
  • Range: (y \in \mathbb{R}).

Applications of Logarithms

• Population Growth: The population of a city grows by a constant rate. [ A = P(1 + i)^n ]
  • If a population triples in size, solve for (n) using [ 3P = P(1 + i)^n ]
• Financial Calculations: Calculate loan repayments and interest rates using logarithms.
• Radioactive Decay: Determine the decay rate using logarithms.
• pH Levels: Use the formula [ \text{pH} = -\log_{10}[\text{H}^+] ]