Chapter 1-5 Section 1 Lesson 1: Understanding Functions and Their Properties

Hello everyone! In this lesson, we're diving deep into the world of functions. We'll explore various types of functions, learn how to evaluate them, and understand how to determine their domains and ranges. Mastering these fundamental concepts is crucial for success in calculus and beyond. So, let's get started!

Topics Covered:

  • Evaluating Functions:
    • Piecewise-Defined Functions: These functions have different rules for different input intervals. For example: $$ f(x) = \begin{cases} x^2 & \text{if } x < 0 \\ 2x + 1 & \text{if } x \geq 0 \end{cases} $$ To evaluate, determine which interval $x$ belongs to and apply the corresponding rule.
    • Absolute Value, Rational, and Radical Functions: Be mindful of potential issues like division by zero (rational functions) or negative values under a square root (radical functions). For example, $f(x) = |x-3|$, $g(x) = \frac{1}{x+2}$, and $h(x) = \sqrt{x-1}$.
  • Graphical Analysis:
    • Finding Outputs and Inputs from a Graph: The graph of a function $y = f(x)$ visually represents input-output pairs. Given an $x$ value, find the corresponding $y$ value on the graph (the output), and vice versa.
    • Identifying Zeroes, Positive, and Negative Intervals: The zeroes are where the graph intersects the x-axis (where $f(x) = 0$). A function is positive where its graph is above the x-axis ($f(x) > 0$), and negative where it's below ($f(x) < 0$).
  • Distance Formula:
    • Distance Between Two Points: The distance $d$ between points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$
  • Domain and Range:
    • From Graphs: The domain is the set of all possible $x$ values, and the range is the set of all possible $y$ values. Identify these by observing the graph's extent along the x and y axes.
      • Continuous Functions: Look for any breaks or asymptotes.
      • Discrete Relations: The domain and range are simply the sets of $x$ and $y$ coordinates, respectively.
      • Piecewise Functions: Consider the domain and range of each piece.
      • Quadratic Functions: The domain is typically all real numbers. The range depends on whether the parabola opens upwards or downwards and the vertex's y-coordinate.
    • Rational Functions: Find excluded values where the denominator equals zero. Express the domain using interval notation. For example, for $f(x) = \frac{1}{x-2}$, the domain is $(-\infty, 2) \cup (2, \infty)$.
    • Square Root Functions: The expression inside the square root must be non-negative. For example, for $g(x) = \sqrt{x+3}$, we need $x+3 \geq 0$, so $x \geq -3$. The domain is $[-3, \infty)$.
    • Fractional Functions Involving Radicals: Identify all restrictions from both the radical and the fraction. For example $h(x) = \frac{\sqrt{x-1}}{x-5}$ requires $x-1 \geq 0$ and $x-5 \neq 0$ so $x \geq 1$ and $x \neq 5$. Domain is $[1, 5) \cup (5, \infty)$.

Remember to practice these concepts with plenty of examples. Understanding the domain and range of a function, for example, is all about knowing the possible inputs and outputs! Keep up the great work, and don't hesitate to ask questions!