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📐 Level 1 - Topic 1: Functions and Their Properties 🚀

The Concept of No Breaks, Jumps, or Holes in a Function's Graph

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1) Introduction to Functions - The Foundation of Calculus

Functions are, without a doubt, a cornerstone of mathematics, and absolutely essential for understanding calculus. They provide a structured way to describe relationships between quantities, allowing us to model real-world phenomena, make predictions, and, ultimately, perform the powerful operations of calculus. Let's delve deeper into what functions truly are.

Definition: Function

A function \( f \) from a set \( A \) (called the domain) to a set \( B \) (called the codomain) is a rule that assigns to each element \( x \) in \( A \) exactly one element \( y \) in \( B \). We denote this assignment as \( f(x) = y \).

Think of a function as a precise machine. You feed it an input (from the domain), it processes it according to a specific, unchanging rule, and spits out a unique output (in the codomain). The crucial aspect is "exactly one output" for each input.

Everyday Examples of Functions

  • Vending Machine: You select a button (input), and it dispenses a specific item (output). Each button corresponds to exactly one item.
  • Postage Rates: The cost to mail a letter (output) depends on its weight (input). For a given weight, there's only one postage cost.
  • Area of a Square: The area of a square (output) is determined by the length of its side (input). For each side length, there’s only one area. Mathematically, \( A(s) = s^2 \).

2) Types of Functions - A Toolkit of Mathematical Relationships

Calculus deals with various types of functions, each with unique properties and behaviors. Understanding these types is crucial for applying calculus techniques effectively. Here are some fundamental categories:

  • Polynomial Functions

    Definition: Polynomial Function

    A polynomial function \( P(x) \) is a function that can be expressed in the form: \[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \] where \( n \) is a non-negative integer (the degree of the polynomial), and \( a_n, a_{n-1}, \dots, a_0 \) are constants called coefficients, with \( a_n \neq 0 \).

    Polynomial functions are characterized by being sums of terms, each of which is a constant coefficient multiplied by a non-negative integer power of \( x \). They are defined for all real numbers.

    Examples of Polynomial Functions

    • Constant Function (Degree 0): \( f(x) = 5 \)
    • Linear Function (Degree 1): \( f(x) = 2x + 3 \)
    • Quadratic Function (Degree 2): \( f(x) = x^2 - 4x + 1 \)
    • Cubic Function (Degree 3): \( f(x) = x^3 + 2x^2 - x + 7 \)
    • Quintic Function (Degree 5): \( f(x) = -3x^5 + x^2 - 9 \)

  • Rational Functions

    Definition: Rational Function

    A rational function \( R(x) \) is a function that can be expressed as the ratio of two polynomial functions, \( P(x) \) and \( Q(x) \): \[ R(x) = \frac{P(x)}{Q(x)} \] where \( P(x) \) and \( Q(x) \) are polynomial functions, and \( Q(x) \neq 0 \).

    Rational functions are defined everywhere except where the denominator polynomial \( Q(x) \) is equal to zero. These points where \( Q(x) = 0 \) lead to vertical asymptotes or holes in the graph of the function.

    Examples of Rational Functions

    • \( f(x) = \frac{1}{x} \)
    • \( g(x) = \frac{x + 2}{x - 3} \)
    • \( h(x) = \frac{x^2 + 1}{x^2 - 4} \)
    • \( R(x) = \frac{3x^4 - x + 5}{x^2 + 2x + 1} \)

  • Exponential Functions

    Definition: Exponential Function

    An exponential function \( E(x) \) is a function of the form: \[ E(x) = a^x \] where \( a \) is a positive constant, \( a > 0 \), and \( a \neq 1 \) (called the base). The variable \( x \) is in the exponent.

    Exponential functions are characterized by rapid growth (if \( a > 1 \)) or decay (if \( 0 < a < 1 \)). They are crucial in modeling phenomena like population growth, radioactive decay, and compound interest.

    Examples of Exponential Functions

    • \( f(x) = 2^x \) (Exponential growth)
    • \( g(x) = \left(\frac{1}{2}\right)^x = 2^{-x} \) (Exponential decay)
    • \( h(x) = e^x \) (Natural exponential function, where \( e \approx 2.71828 \))
    • \( E(x) = 10^{x+1} \)

  • Logarithmic Functions

    Definition: Logarithmic Function

    A logarithmic function \( L(x) \) is the inverse function of an exponential function. It is typically written as: \[ L(x) = \log_a(x) \] where \( a \) is the base, \( a > 0 \) and \( a \neq 1 \). For calculus, the most important base is \( e \), giving the natural logarithm, \( \ln(x) = \log_e(x) \).

    Logarithmic functions "undo" exponentiation. If \( y = a^x \), then \( x = \log_a(y) \). Logarithmic functions are defined only for positive inputs \( x > 0 \). They are used to model phenomena on a logarithmic scale and are essential in solving exponential equations.

    Examples of Logarithmic Functions

    • \( f(x) = \log_{10}(x) \) (Common logarithm)
    • \( g(x) = \ln(x) \) (Natural logarithm)
    • \( h(x) = \log_2(x + 5) \)
    • \( L(x) = -\ln(x^2) \)

  • Trigonometric Functions

    Definition: Trigonometric Functions

    Trigonometric functions relate angles of a right triangle to ratios of its sides. The primary trigonometric functions are sine (\( \sin(x) \)), cosine (\( \cos(x) \)), and tangent (\( \tan(x) \)). Others include cosecant (\( \csc(x) \)), secant (\( \sec(x) \)), and cotangent (\( \cot(x) \)).

    These functions are periodic and are fundamental in modeling periodic phenomena like waves, oscillations, and rotations. They are typically introduced in trigonometry but are essential in calculus as well. We assume you have some familiarity with these from previous studies.

    Examples of Trigonometric Functions

    • \( f(x) = \sin(x) \) (Sine function)
    • \( g(x) = \cos(x) \) (Cosine function)
    • \( h(x) = \tan(x) \) (Tangent function)
    • \( T(x) = 2\sin(3x) - \cos(x/2) \) (Combination of trigonometric functions)

3) Domain and Range - Input and Output Boundaries

Understanding the domain and range of a function is critical. They define the allowable inputs and the possible outputs of a function, setting the stage for meaningful mathematical operations and interpretations in calculus.

\[ \text{Domain: Set of all possible input values } x \text{ for which } f(x) \text{ is defined.} \] \[ \text{Range: Set of all possible output values } y = f(x) \text{ that the function produces.} \]

3.1) Determining the Domain - Finding Valid Inputs

To find the domain of a function, we need to identify any values of \( x \) for which the function is not defined in the real number system. Common restrictions come from:

  • Denominators cannot be zero: Exclude any \( x \) that makes a denominator zero.
  • Even roots of negative numbers are not real: For functions with square roots, fourth roots, etc., the expression under the root must be non-negative (\( \geq 0 \)).
  • Logarithms are only defined for positive arguments: For logarithmic functions, the argument must be strictly positive (\( > 0 \)).

Examples: Finding Domains

  1. Example 1: Find the domain of \( f(x) = \frac{3x}{x^2 - 4} \).

    Solution: The denominator is \( x^2 - 4 \). We need to find values of \( x \) for which \( x^2 - 4 = 0 \). \( x^2 - 4 = (x - 2)(x + 2) = 0 \). So, \( x = 2 \) or \( x = -2 \). These values must be excluded. Domain: \( \{x \in \mathbb{R} \mid x \neq 2, x \neq -2 \} \) or in interval notation, \( (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \).

  2. Example 2: Find the domain of \( g(x) = \sqrt{5 - x} \).

    Solution: We need the expression under the square root to be non-negative: \( 5 - x \geq 0 \). Solving for \( x \), we get \( x \leq 5 \). Domain: \( \{x \in \mathbb{R} \mid x \leq 5 \} \) or in interval notation, \( (-\infty, 5] \).

  3. Example 3: Find the domain of \( h(x) = \ln(x^2 + 1) \).

    Solution: For the natural logarithm, we need \( x^2 + 1 > 0 \). Since \( x^2 \) is always non-negative (\( x^2 \geq 0 \)), \( x^2 + 1 \) will always be greater than 0 for any real number \( x \). Domain: \( \{x \in \mathbb{R} \} \) or \( (-\infty, \infty) \).

3.2) Determining the Range - Finding Possible Outputs

Finding the range involves determining all possible \( y \)-values that the function \( f(x) \) can produce. This can be more challenging than finding the domain, but there are several techniques.

  • Graphical Method: Visualize the graph of \( y = f(x) \). The range is the set of all \( y \)-coordinates covered by the graph.
  • Algebraic Method: Try to solve \( y = f(x) \) for \( x \) in terms of \( y \). Then determine the values of \( y \) for which \( x \) is a real number and in the domain of \( f \).
  • Using Function Properties: Utilize properties of known functions and transformations to deduce the range.

Examples: Finding Ranges

  1. Example 1: Find the range of \( f(x) = x^2 - 3 \).

    Solution (Graphical & Algebraic): The graph of \( y = x^2 - 3 \) is a parabola opening upwards, vertex at \( (0, -3) \). Graphically, the lowest \( y \) value is \( -3 \), and it extends upwards. Algebraically, since \( x^2 \geq 0 \), \( x^2 - 3 \geq -3 \). For any \( y \geq -3 \), we can solve \( y = x^2 - 3 \) for real \( x \) as \( x = \pm \sqrt{y + 3} \). Range: \( \{y \in \mathbb{R} \mid y \geq -3 \} \) or \( [-3, \infty) \).

  2. Example 2: Find the range of \( g(x) = \frac{1}{x^2 + 1} \).

    Solution (Algebraic & Function Properties): Let \( y = \frac{1}{x^2 + 1} \). We know \( x^2 \geq 0 \), so \( x^2 + 1 \geq 1 \). Thus, \( 0 < \frac{1}{x^2 + 1} \leq 1 \). Also, \( x^2 + 1 \) can be any value from 1 to \( \infty \). As \( x^2 + 1 \) gets very large, \( \frac{1}{x^2 + 1} \) approaches 0 but never reaches it. The maximum value is when \( x^2 + 1 = 1 \) (at \( x = 0 \)), giving \( y = 1 \). Range: \( \{y \in \mathbb{R} \mid 0 < y \leq 1 \} \) or \( (0, 1] \).

  3. Example 3: Find the range of \( h(x) = 2\sin(x) \).

    Solution (Function Properties): We know the range of \( \sin(x) \) is \( [-1, 1] \). Multiplying by 2, we stretch the range vertically by a factor of 2. Range of \( 2\sin(x) \) is \( [2 \times (-1), 2 \times 1] = [-2, 2] \).

4) Transformations of Functions - Modifying Graphs and Equations

Transformations allow us to manipulate the graph of a function, and consequently, its equation. Understanding transformations is crucial for graphing and analyzing functions.

  • 4.1) Vertical Shifts

    Adding a constant \( c \) to a function, \( f(x) + c \), shifts the graph vertically.

    • If \( c > 0 \), the graph shifts upwards by \( c \) units.
    • If \( c < 0 \), the graph shifts downwards by \( |c| \) units.
    \[ y = f(x) + c \]

    Example: Vertical Shifts

    Consider \( f(x) = x^2 \).

    • \( g(x) = x^2 + 2 = f(x) + 2 \) shifts the graph of \( f(x) \) up by 2 units.
    • \( h(x) = x^2 - 3 = f(x) - 3 \) shifts the graph of \( f(x) \) down by 3 units.

  • 4.2) Horizontal Shifts

    Replacing \( x \) with \( (x - c) \) in a function, \( f(x - c) \), shifts the graph horizontally. Note the sign change!

    • If \( c > 0 \), the graph shifts to the right by \( c \) units.
    • If \( c < 0 \), the graph shifts to the left by \( |c| \) units.
    \[ y = f(x - c) \]

    Example: Horizontal Shifts

    Consider \( f(x) = \sqrt{x} \).

    • \( g(x) = \sqrt{x - 2} = f(x - 2) \) shifts the graph of \( f(x) \) right by 2 units.
    • \( h(x) = \sqrt{x + 3} = \sqrt{x - (-3)} = f(x - (-3)) \) shifts the graph of \( f(x) \) left by 3 units.

  • 4.3) Reflections

    Reflections flip the graph of a function across an axis.

    • Reflection across the x-axis: Multiply the function by \( -1 \), \( -f(x) \). This flips the graph over the x-axis.
    • Reflection across the y-axis: Replace \( x \) with \( -x \), \( f(-x) \). This flips the graph over the y-axis.
    \[ y = -f(x) \quad \text{(x-axis reflection)} \] \[ y = f(-x) \quad \text{(y-axis reflection)} \]

    Examples: Reflections

    Consider \( f(x) = e^x \).

    • \( g(x) = -e^x = -f(x) \) reflects the graph of \( f(x) \) over the x-axis.
    • \( h(x) = e^{-x} = f(-x) \) reflects the graph of \( f(x) \) over the y-axis.

  • 4.4) Vertical Stretching and Shrinking

    Multiplying a function by a constant \( a \) vertically stretches or shrinks the graph.

    • If \( a > 1 \), \( a f(x) \) vertically stretches the graph of \( f(x) \) by a factor of \( a \).
    • If \( 0 < a < 1 \), \( a f(x) \) vertically shrinks (compresses) the graph of \( f(x) \) by a factor of \( a \).
    • If \( a < 0 \), it involves a vertical stretch/shrink and a reflection across the x-axis.
    \[ y = a f(x) \]

    Examples: Vertical Stretching/Shrinking

    Consider \( f(x) = \sin(x) \).

    • \( g(x) = 2\sin(x) = 2f(x) \) vertically stretches the graph of \( f(x) \) by a factor of 2. The range becomes \( [-2, 2] \) instead of \( [-1, 1] \).
    • \( h(x) = \frac{1}{2}\sin(x) = \frac{1}{2}f(x) \) vertically shrinks the graph of \( f(x) \) by a factor of 1/2. The range becomes \( [-1/2, 1/2] \).

  • 4.5) Horizontal Stretching and Shrinking

    Replacing \( x \) with \( bx \) in a function, \( f(bx) \), horizontally stretches or shrinks the graph. Note the inverse effect on the horizontal scale!

    • If \( b > 1 \), \( f(bx) \) horizontally shrinks (compresses) the graph of \( f(x) \) by a factor of \( \frac{1}{b} \).
    • If \( 0 < b < 1 \), \( f(bx) \) horizontally stretches the graph of \( f(x) \) by a factor of \( \frac{1}{b} \).
    \[ y = f(bx) \]

    Examples: Horizontal Stretching/Shrinking

    Consider \( f(x) = \cos(x) \). The period of \( \cos(x) \) is \( 2\pi \).

    • \( g(x) = \cos(2x) = f(2x) \) horizontally shrinks the graph of \( f(x) \) by a factor of \( \frac{1}{2} \). The period of \( \cos(2x) \) becomes \( \pi \).
    • \( h(x) = \cos\left(\frac{1}{2}x\right) = f\left(\frac{1}{2}x\right) \) horizontally stretches the graph of \( f(x) \) by a factor of \( \frac{1}{1/2} = 2 \). The period of \( \cos\left(\frac{1}{2}x\right) \) becomes \( 4\pi \).

  • 4.6) Combining Transformations

    Functions can undergo multiple transformations. The order in which transformations are applied can matter. A general guideline is to follow this order:

    1. Horizontal shifts
    2. Stretching or shrinking (horizontal and vertical)
    3. Reflections
    4. Vertical shifts

    Example: Combined Transformations

    Describe the transformations needed to go from \( f(x) = x^2 \) to \( g(x) = -2(x + 1)^2 + 3 \).

    1. Start with \( y = x^2 \).
    2. Horizontal shift left by 1 unit: Replace \( x \) with \( (x + 1) \) to get \( y = (x + 1)^2 \).
    3. Vertical stretch by a factor of 2 and reflection across x-axis: Multiply by \( -2 \) to get \( y = -2(x + 1)^2 \).
    4. Vertical shift up by 3 units: Add 3 to get \( y = -2(x + 1)^2 + 3 \).

5) Inverse Functions - Undoing Function Operations

An inverse function, denoted as \( f^{-1} \), reverses the operation of a function \( f \). If \( f \) takes an input \( x \) to an output \( y \), then \( f^{-1} \) takes \( y \) back to \( x \). Not all functions have inverses that are also functions. For a function to have an inverse function, it must be one-to-one.

Definition: Inverse Function

If \( f \) is a one-to-one function with domain \( A \) and range \( B \), then its inverse function \( f^{-1} \) has domain \( B \) and range \( A \), and is defined by: \[ \text{if } f(x) = y \text{, then } f^{-1}(y) = x \]

This means \( f^{-1} \) "undoes" what \( f \) does. In terms of composition:

\[ f^{-1}(f(x)) = x \quad \text{for all } x \text{ in the domain of } f \] \[ f(f^{-1}(y)) = y \quad \text{for all } y \text{ in the range of } f \]

5.1) How to Find the Inverse Function Algebraically

To find the inverse function of a one-to-one function \( y = f(x) \), we typically follow these steps:

  1. Replace \( f(x) \) with \( y \): Write down the equation \( y = f(x) \).
  2. Swap \( x \) and \( y \): Interchange \( x \) and \( y \) to get \( x = f(y) \). This step reflects the reversal of input and output.
  3. Solve for \( y \) in terms of \( x \): If possible, solve the equation \( x = f(y) \) for \( y \). The resulting expression for \( y \) in terms of \( x \) is the inverse function \( f^{-1}(x) \).
  4. Replace \( y \) with \( f^{-1}(x) \): Denote the solved \( y \) as \( f^{-1}(x) \).
  5. Verify (optional but recommended): Check if \( f^{-1}(f(x)) = x \) and \( f(f^{-1}(x)) = x \) to confirm that you have found the correct inverse.

Examples: Finding Inverse Functions

  1. Example 1: Find the inverse of \( f(x) = 2x + 3 \).

    Solution:

    1. Let \( y = 2x + 3 \).
    2. Swap \( x \) and \( y \): \( x = 2y + 3 \).
    3. Solve for \( y \): \( x - 3 = 2y \Rightarrow y = \frac{x - 3}{2} \).
    4. Replace \( y \) with \( f^{-1}(x) \): \( f^{-1}(x) = \frac{x - 3}{2} \).
    5. Verification:
      • \( f^{-1}(f(x)) = f^{-1}(2x + 3) = \frac{(2x + 3) - 3}{2} = \frac{2x}{2} = x \).
      • \( f(f^{-1}(x)) = f\left(\frac{x - 3}{2}\right) = 2\left(\frac{x - 3}{2}\right) + 3 = (x - 3) + 3 = x \).
      Verification holds.
    Thus, the inverse function is \( f^{-1}(x) = \frac{x - 3}{2} \).

  2. Example 2: Find the inverse of \( g(x) = \frac{x}{x - 1} \).

    Solution:

    1. Let \( y = \frac{x}{x - 1} \).
    2. Swap \( x \) and \( y \): \( x = \frac{y}{y - 1} \).
    3. Solve for \( y \): \( x(y - 1) = y \Rightarrow xy - x = y \Rightarrow xy - y = x \Rightarrow y(x - 1) = x \Rightarrow y = \frac{x}{x - 1} \).
    4. Replace \( y \) with \( g^{-1}(x) \): \( g^{-1}(x) = \frac{x}{x - 1} \). In this case, the inverse function is the same as the original function!
    5. Verification: (You can verify similarly as in Example 1).
    Thus, the inverse function is \( g^{-1}(x) = \frac{x}{x - 1} \).

5.2) When Does a Function Have an Inverse? - One-to-One Functions and Horizontal Line Test

Not every function has an inverse function. For a function to have an inverse function, it must be one-to-one, also called injective.

Definition: One-to-One Function

A function \( f \) is one-to-one if it never takes on the same value twice; that is, \[ \text{if } x_1 \neq x_2 \text{, then } f(x_1) \neq f(x_2) \] Equivalently, if \( f(x_1) = f(x_2) \), then \( x_1 = x_2 \).

Graphically, we can check if a function is one-to-one using the Horizontal Line Test:

Horizontal Line Test:

A function is one-to-one if and only if no horizontal line intersects its graph more than once.

If a horizontal line intersects the graph more than once, it means there are at least two different \( x \)-values that give the same \( y \)-value, so the function is not one-to-one and does not have an inverse function (over its entire domain).

Examples: One-to-One and Not One-to-One Functions

  1. Example 1: \( f(x) = x^2 \). Not one-to-one.

    The parabola \( y = x^2 \) fails the horizontal line test. For instance, the horizontal line \( y = 4 \) intersects the parabola at \( x = 2 \) and \( x = -2 \). Since \( f(2) = f(-2) = 4 \) but \( 2 \neq -2 \), \( f(x) = x^2 \) is not one-to-one and does not have an inverse function over its entire domain \( (-\infty, \infty) \). However, if we restrict the domain to \( [0, \infty) \) or \( (-\infty, 0] \), then \( f(x) = x^2 \) *does* become one-to-one and has an inverse ( \( f^{-1}(x) = \sqrt{x} \) for domain \( [0, \infty) \) and \( f^{-1}(x) = -\sqrt{x} \) for domain \( (-\infty, 0] \)).

  2. Example 2: \( g(x) = x^3 \). One-to-one.

    The graph of \( y = x^3 \) passes the horizontal line test. Any horizontal line will intersect the cubic curve at most once. Algebraically, if \( x_1^3 = x_2^3 \), then taking the cube root of both sides gives \( x_1 = x_2 \). So, \( g(x) = x^3 \) is one-to-one and has an inverse function \( g^{-1}(x) = \sqrt[3]{x} \).

  3. Example 3: \( h(x) = \sin(x) \). Not one-to-one.

    The sine function \( y = \sin(x) \) is periodic. Many horizontal lines (e.g., \( y = 0.5 \)) will intersect the sine curve infinitely many times. Thus, \( \sin(x) \) is not one-to-one over its entire domain \( (-\infty, \infty) \) and does not have an inverse function over its entire domain. To define an inverse sine function (arcsin or \( \sin^{-1} \)), we must restrict the domain of \( \sin(x) \) to an interval where it *is* one-to-one, conventionally \( [-\frac{\pi}{2}, \frac{\pi}{2}] \). On this restricted domain, sine does have an inverse function.

6) Practice Questions - Test Your Understanding

Now it's time to practice what you've learned! Below are fundamental and challenging questions to help solidify your understanding of functions and their properties.

Fundamental Practice Questions

Instructions: Solve the following problems to reinforce your basic understanding of functions, domain, range, and transformations.

  1. Q1. Determine whether the relation described by \( y = \pm\sqrt{x} \) is a function of \( x \). Explain your reasoning.
  2. Q2. For the function \( f(x) = \frac{2x - 6}{x + 2} \), identify its domain.
  3. Q3. Find the range of the function \( g(x) = x^2 + 5 \) for \( x \in [-2, 3] \).
  4. Q4. Describe the transformations applied to \( f(x) = |x| \) to obtain \( h(x) = -|x - 3| + 1 \).
  5. Q5. Determine if the function \( k(x) = \frac{1}{x^2 + 1} \) has a horizontal asymptote. If so, what is it?
  6. Q6. Find the inverse function of \( f(x) = 4x - 7 \).
  7. Q7. Is the function \( p(x) = x^3 + x \) one-to-one? Justify your answer. (Hint: Consider its derivative or graph behavior, though derivatives are formally introduced later, you can think about whether \(x^3+x\) is always increasing or decreasing).
  8. Q8. What is the domain of the composite function \( (f \circ g)(x) \) if \( f(x) = \sqrt{x} \) and \( g(x) = 4 - x \)?
  9. Q9. Sketch the graph of \( y = -(x - 2)^2 + 1 \) by starting with \( y = x^2 \) and applying transformations.
  10. Q10. Find the domain and range of \( q(x) = 3e^{x} - 2 \).

Challenging Practice Questions

Instructions: These questions require deeper thinking and application of the concepts. Solve them to further enhance your understanding.

  1. Q1. Find the domain of \( F(x) = \sqrt{\frac{x + 2}{3 - x}} \). Express your answer in interval notation.
  2. Q2. Determine the range of \( G(x) = \frac{2}{x^2 + 2x + 2} \). (Hint: Complete the square in the denominator).
  3. Q3. A function \( f(x) \) is transformed by shifting it 2 units to the right, reflecting it across the y-axis, and then stretching it vertically by a factor of 3. If the original function was \( f(x) = \ln(x) \), find the equation of the transformed function \( H(x) \).
  4. Q4. For what value(s) of \( c \) does the function \( f(x) = \frac{cx}{x + c} \) equal its own inverse?
  5. Q5. Prove algebraically that \( f(x) = \frac{3x + 1}{x - 2} \) is one-to-one on its domain. Then, find its inverse function and the domain and range of \( f^{-1}(x) \).

7) Summary & Cheat Sheet - Key Concepts at a Glance

Let's summarize the key concepts covered in this topic. This cheat sheet is designed to help you quickly recall the essential ideas about functions and their properties.

7.1) Core Function Concepts

  • Function Definition: A rule assigning exactly one output for each input.
  • Domain: Set of all valid inputs (x-values). Consider restrictions: denominators, even roots, logarithms.
  • Range: Set of all possible outputs (y-values). Found graphically, algebraically, or by analyzing function behavior.
  • Function Notation: \( y = f(x) \), \( f \) is the function name, \( x \) is input, \( f(x) \) is output.
  • Types of Functions: Polynomial, Rational, Exponential, Logarithmic, Trigonometric – know their basic forms and properties.

7.2) Transformations of Functions - How Graphs Change

  • Vertical Shifts: \( f(x) + c \) (up if \( c > 0 \), down if \( c < 0 \)).
  • Horizontal Shifts: \( f(x - c) \) (right if \( c > 0 \), left if \( c < 0 \)).
  • Reflections: \( -f(x) \) (x-axis), \( f(-x) \) (y-axis).
  • Vertical Stretch/Shrink: \( a f(x) \) (stretch if \( |a| > 1 \), shrink if \( 0 < |a| < 1 \)).
  • Horizontal Stretch/Shrink: \( f(bx) \) (shrink if \( |b| > 1 \), stretch if \( 0 < |b| < 1 \)).
  • Order of Transformations: Horizontal, Stretch/Shrink, Reflect, Vertical.

7.3) Inverse Functions - Undoing Operations

  • Inverse Function \( f^{-1} \): Reverses the action of \( f \). If \( f(x) = y \), then \( f^{-1}(y) = x \).
  • Finding \( f^{-1}(x) \) Algebraically: Swap \( x \) and \( y \) in \( y = f(x) \), then solve for \( y \).
  • Condition for Inverse Function: Function must be one-to-one (pass the Horizontal Line Test).
  • Verification: \( f^{-1}(f(x)) = x \) and \( f(f^{-1}(x)) = x \).

Graph Functions on Desmos ↗

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