Understanding Functions

Definition of a Function

• A function is a relationship between input and output.
• Relates each input to a single output.
• For a set of inputs and outputs, each input should have one output for it to be a function.
• Example of functions:
  • $f(x) = 3x + 5$
  • $y = 2x - 7$ (linear function)
• Input represents the domain; output represents the range.

Properties of Functions

• Single Output Rule:
  • Each input should give a single output.
  • If an input gives multiple outputs, it's not a function but a relation.
• Function Tables:
  • Example: $y = 2x + 4$
    • If $x = 1$, $y = 6$
    • If $x = 2$, $y = 8$
    • If $x = 3$, $y = 10$
  • Function tables should have unique x values leading to unique y values.

Identifying Functions

• Vertical Line Test:
  • If a vertical line intersects the graph in more than one point, it's not a function.
• Identifying from Graphs:
  • Graph A: Function
  • Graph B: Not a function
  • Graph C: Function
  • Graph D: Not a function
  • Graph E: Function
  • Graph F: Not a function

Types of Functions

• Linear Functions:
  • Straight line, e.g., $y = 2x + 7$
• Quadratic Functions:
  • Parabola, e.g., $y = x^2 + 4$
• Cubic Functions:
  • Examples include $y = x^3$
• Radical Functions:
  • Square root of x, e.g., $y = \sqrt{x}$
• Logarithmic Functions:
  • e.g., $y = \log(x)$
• Exponential Functions:
  • e.g., $y = e^x$, $e \approx 2.718$
• Trigonometric Functions:
  • e.g., $y = \sin(x)$
• Absolute Value Functions:
  • e.g., $y = |x|$
• Rational Functions:
  • e.g., $y = \frac{1}{x}$
• Polynomial Functions:
  • Many terms, e.g., $y = x^4 - x^3 + 2x^2 - 7$

Evaluating Functions

• Single Variable Example:
  • $f(x) = x^2 + 4x - 7$
  • Evaluate $f(3)$: $3^2 + 4(3) - 7 = 14$
• Multivariable Functions:
  • $f(x, y) = 2x^2 - y^2 + 3xy$
  • Evaluate $f(2, 3)$: $2(2)^2 - 3^2 + 3(2)(3) = 17$

Additional Resources

• Mention of additional examples and problems available online for further learning.