Functional relationship (investigate functions)

Understanding Functional Relationships

A functional relationship is a connection between two sets, typically referred to as the domain and the codomain, where each element of the domain is associated with exactly one element of the codomain. This relationship is often expressed as a function. Investigating functions involves understanding their properties, behaviors, and the ways in which they map elements from the domain to the codomain.

Key Concepts of Functions

  • Domain: The set of all possible inputs for the function.
  • Codomain: The set of all possible outputs that the function can produce.
  • Range: The set of all actual outputs that the function produces.
  • Mapping: The process of associating each element of the domain with an element in the codomain.

Types of Functions

Type of Function Definition Example
Linear Function A function with a constant rate of change, often in the form $f(x) = mx + b$. $f(x) = 2x + 3$
Quadratic Function A function that can be written in the form $f(x) = ax^2 + bx + c$. $f(x) = x^2 - 4x + 4$
Exponential Function A function where the variable is in the exponent, typically in the form $f(x) = a \cdot b^x$. $f(x) = 3 \cdot 2^x$
Logarithmic Function The inverse of an exponential function, often in the form $f(x) = \log_b(x)$. $f(x) = \log_2(x)$
Trigonometric Function Functions that relate angles of a triangle to the lengths of its sides. $f(x) = \sin(x)$

Investigating Functions

When investigating functions, there are several key aspects to consider:

  1. Domain and Range: Determine the set of all possible inputs (domain) and the set of all actual outputs (range).

  2. Intercepts: Find where the function crosses the x-axis (x-intercepts) and y-axis (y-intercepts).

  3. Asymptotes: Identify any lines that the function approaches but never touches.

  4. Symmetry: Check if the function is even, odd, or neither.

  5. Continuity: Determine if the function has any breaks, holes, or discontinuities.

  6. End Behavior: Analyze what happens to the function values as $x$ approaches infinity or negative infinity.

  7. Intervals of Increase/Decrease: Identify where the function is going up or down as $x$ increases.

  8. Relative Maximums/Minimums: Find local high and low points of the function.

Formulas and Properties

  • Linear Function: $f(x) = mx + b$

    • Slope ($m$) indicates the rate of change.
    • Y-intercept ($b$) is where the function crosses the y-axis.

  • Quadratic Function: $f(x) = ax^2 + bx + c$

    • The vertex form is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex.
    • The axis of symmetry is the vertical line $x = h$.

  • Exponential Function: $f(x) = a \cdot b^x$

    • The base ($b$) determines the growth or decay rate.
    • The function has a horizontal asymptote at $y = 0$.

  • Logarithmic Function: $f(x) = \log_b(x)$

    • The inverse of the exponential function $y = b^x$.
    • The function has a vertical asymptote at $x = 0$.

  • Trigonometric Function: $f(x) = \sin(x)$, $f(x) = \cos(x)$, $f(x) = \tan(x)$, etc.

    • Periodic functions with specific periods and amplitudes.
    • Often have symmetries about the origin or the y-axis.

Examples

  1. Linear Function Example:

    • Consider $f(x) = 2x + 3$.
    • Domain: All real numbers.
    • Range: All real numbers.
    • Y-intercept: $(0, 3)$.
    • X-intercept: Solve $0 = 2x + 3$ to find $x = -\frac{3}{2}$.

  2. Quadratic Function Example:

    • Take $f(x) = x^2 - 4x + 4$.
    • Domain: All real numbers.
    • Range: $y \geq 0$ (since the parabola opens upwards).
    • Vertex: $(2, 0)$, found by completing the square or using the formula $h = -\frac{b}{2a}$.
    • Axis of symmetry: $x = 2$.

  3. Exponential Function Example:

    • Consider $f(x) = 3 \cdot 2^x$.
    • Domain: All real numbers.
    • Range: $y > 0$ (since the base is positive and greater than 1).
    • Y-intercept: $(0, 3)$.
    • No x-intercepts (the function never crosses the x-axis).

  4. Logarithmic Function Example:

    • Take $f(x) = \log_2(x)$.
    • Domain: $x > 0$ (logarithms are undefined for non-positive numbers).
    • Range: All real numbers.
    • X-intercept: $(1, 0)$ (since $\log_2(1) = 0$).
    • Vertical asymptote: $x = 0$.

  5. Trigonometric Function Example:

    • Consider $f(x) = \sin(x)$.
    • Domain: All real numbers.
    • Range: $-1 \leq y \leq 1$.
    • Period: $2\pi$.
    • Symmetry: Origin symmetry (odd function).

When studying for exams, it's crucial to practice identifying these properties and characteristics for various functions. Understanding the functional relationship and being able to investigate functions is a foundational skill in mathematics that applies to many areas, including calculus, physics, engineering, and economics.