Section 2.3: Properties of Functions

Welcome back to Professor Baker's Math Class! In Section 2.3, we'll be exploring the fundamental properties of functions, which are essential for understanding and working with mathematical models. This section will cover domain, range, intercepts, symmetry, and more. Let's get started!

Understanding Domain and Range

The domain of a function is the set of all possible input values (often $x$-values) for which the function is defined. The range, on the other hand, is the set of all possible output values (often $y$-values) that the function can produce.

To find the domain, consider any restrictions on the input values. For example:

  • Division by zero: Avoid values that make the denominator of a fraction equal to zero.
  • Square roots of negative numbers: Ensure that the expression inside a square root is non-negative.
  • Logarithms of non-positive numbers: Ensure that the argument of a logarithm is positive.

The range can be found by considering the possible output values based on the function's behavior. Sometimes graphing the function is helpful to visualize the range.

Example: Consider the function $f(x) = \sqrt{x-2}$. The domain is $x \geq 2$ because the expression inside the square root must be non-negative. The range is $f(x) \geq 0$ because the square root function always returns a non-negative value.

Intercepts: Where Functions Meet the Axes

Intercepts are the points where the graph of a function intersects the $x$-axis and the $y$-axis.

  • $x$-intercepts (or roots): Occur when $f(x) = 0$. To find them, set the function equal to zero and solve for $x$.
  • $y$-intercept: Occurs when $x = 0$. To find it, evaluate $f(0)$.

Example: For the function $f(x) = x^2 - 4$, the $x$-intercepts are $x = 2$ and $x = -2$ (because $f(2) = 0$ and $f(-2) = 0$). The $y$-intercept is $f(0) = -4$.

Symmetry: Recognizing Patterns

Symmetry helps us understand the behavior of functions and makes graphing easier. There are three main types of symmetry to consider:

  • Even functions (Symmetric about the y-axis): $f(-x) = f(x)$ for all $x$ in the domain. Example: $f(x) = x^2$
  • Odd functions (Symmetric about the origin): $f(-x) = -f(x)$ for all $x$ in the domain. Example: $f(x) = x^3$
  • Functions with no symmetry: Most functions do not exhibit any symmetry.

To test for symmetry, replace $x$ with $-x$ in the function and simplify. Compare the result to the original function to determine if it's even, odd, or neither.

Piecewise-Defined Functions

A piecewise-defined function is a function defined by multiple sub-functions, each applying to a specific interval of the domain. It's like having different rules for different values of x.

For example:

$$ f(x) = \begin{cases} x^2, & x < 0 \\ 1, & 0 \leq x \leq 2 \\ x - 1, & x > 2 \end{cases} $$

When evaluating a piecewise function, it's crucial to determine which sub-function applies based on the value of $x$.

Putting It All Together

Understanding these properties – domain, range, intercepts, and symmetry – is crucial for analyzing functions and their graphs. Keep practicing, and you'll become more comfortable with identifying and applying these concepts. Good luck, and see you in the next section!