Ｃａｌｃｕｌｕｓ　１　Ｌｅｃｔｕｒｅ　０．２

Ｉｎｔｒｏｄｕｃｔｉｏｎ　ｔｏ　Ｆｕｎｃｔｉｏｎｓ

Ｉｎｔｒｏｄｕｃｔｉｏｎ

• [00:01] Introduction – [📷image]
  • Presentation of the topic: Functions (Section 0.2)
  • Initial definition of a function: one variable depends on another (e.g., $𝒚$ depends on $𝒙$)
  • Fundamental condition for a function: each input has exactly one output
    • If one input had multiple outputs, it would not be a function
  • Common notation for inputs ($𝒙$) and outputs ($𝒚$ or $𝒇(𝒙)$)
  • Ways to represent functions
    • Tables
      • [02:00] 🧩 Example – Function table: Fish caught and their weight
        • Clarification about having specific outputs for each input
        • It is allowed for different inputs to share the same output
      • [05:25] Reference table for a function:
        • $\displaystyle \begin{array}{c|cccc} x & 0 & 1 & 2 & 3 \\ \hline y & 2 & 5 & -3 & 9 \end{array}$
      • [05:50] Reference table for not a function.
        • $\displaystyle \begin{array}{c|cccc} x & 0 & 1 & 2 & 3 & 2\\ \hline y & 2 & 5 & -3 & 9 & 7 \end{array}$
      • [07:05] 🧩 Example – if $𝒚=𝒇(𝒙)$, find $𝒇(0)$
        • Reference table for examples:
          • $\displaystyle \begin{array}{c|cccc} x & 0 & 1 & 2 & 3 \\ \hline y & 2 & 5 & -3 & 9 \end{array}$
        • Interpreting $𝒇(0)$ as the output when the input is 0
          • $𝒇(0)=2$
        • Another example: find $𝒇(3)$
          • $𝒇(3)=9$
    • Formulas
    • Graphs

Ｆｕｎｃｔｉｏｎ　ｎｏｔａｔｉｏｎ

• [07:05] Typical application with equations (e.g., $𝒚=3𝒙^{2}-4𝒙+2$) – [📷image]
  • Isn't the best way to represent a function because when we are looking at two different functions at once, we are not going to distinguish between them.
• [08:24] Advantages of using $𝒇(𝒙)$, $𝓰(𝒙)$, $𝒉(𝒙)$
  • Helps distinguish between multiple functions
  • 🧩 Example – Find $𝒇(0)$ for $𝒇(𝒙)=3𝒙^{2}-4𝒙+2$
  • Contrast $𝒇(0)=2$ with $𝒚=2$
    • $𝒇(0)$ inherently shows the value of the input
    • We obtain also a coordinate point $(0,2)$
      • Coordinate points expressed as $(𝒙,𝒇(𝒙))$

Ｉｄｅｎｔｉｆｙｉｎｇ　ｆｕｎｃｔｉｏｎｓ　ｆｒｏｍ　ｇｒａｐｈｓ

• [09:40] The Vertical Line Test – [📷image]
  • Reminder: in tables, repeated inputs with different outputs mean it is not a function
  • Explanation of the Vertical Line Test
    • A graph is a function if and only if every vertical line intersects it at most once
  • Graphic examples
    • A diagonal line: it is a function
    • A parabola: it is a function (though not one-to-one)
    • A graph that passes the test even if it has an empty section (e.g., $1/𝒙$)
      • Not every input must have an output, but if it does, it must be unique
    • A graph that fails the Vertical Line Test (one input with multiple outputs) is not a function
• [13:20] 🧩 Example – Equation that does not represent a function: the circle – [📷image]
  • Equation of a circle centered at the origin: $𝒙^{2}+𝒚^{2}=25$
  • When trying to solve for $𝒚$, we get $𝒚=\pm\sqrt{25-𝒙^{2}}$
  • The $\pm$ indicates one input ($𝒙$) can have two different outputs ($𝒚$), so it is not a function
  • However, parts of the circle can be functions if defined separately
    • $𝒇(𝒙)=\sqrt{25-𝒙^{2}}$ (the upper half)
    • $𝓰(𝒙)=-\sqrt{25-𝒙^{2}}$ (the lower half)

Ｐｉｅｃｅｗｉｓｅ　Ｆｕｎｃｔｉｏｎｓ

• [17:40] General idea – [📷image]
  • The function’s formula depends on the value of $𝒙$
    • The function changes depending on the input value
• [19:10] Defining absolute value as an example of a piecewise function – [📷image]
  • The absolute value symbol: $|𝒙|$
  • Intuitive definition: distance from zero
  • More specific two-part definition:
    • If $𝒙\ge 0$, $|𝒙|=𝒙$
    • If $𝒙<0$, $|𝒙|=-𝒙$ (the sign is flipped)
  • 🧩 Example – $|5|=5$; $|-12|=12$
• [20:45] Formal notation for a piecewise function – [📷image]
  • Use of braces to group the different definitions and their $𝒙$-ranges
  • 🧩 Example of the piecewise definition for absolute value
• [23:50] Graphing piecewise functions
  • Each “piece” is graphed individually
  • The correct domain interval must be applied to each piece
  • [25:00] Graphing absolute value function – [📷image]
  • [27:35] 🧩 Example – Graphing a piecewise function – [📷image]
    • $\displaystyle 𝒇(𝒙)=\begin{cases}0,& \text{if } 𝒙\le -1,\\\sqrt{1-𝒙^{2}},& -1 <𝒙<1,\\𝒙,& \text{if } 𝒙\ge 1\end{cases}$
    • Identifying key intervals where the function’s definition changes
    • $𝒇(𝒙)=0$ for $𝒙\le−1$
      • Graph of $𝒚=0$ (the x-axis) for $𝒙\le−1$
      • Closed circle at $𝒙=−1$ because of “≤”
    • $𝒇(𝒙)=\sqrt{1−𝒙^{2}}$ for $−1<𝒙<1$
      • Rewriting as $𝒚=\sqrt{1−𝒙^{2}}$
      • Recognizing the related circle equation ($𝒙^{2}+𝒚^{2}=1$)
      • Identifying it as the upper semicircle of radius 1, centered at (0, 0)
      • Open circles because of the strict inequality “<”
        • At $𝒙=−1$
          • Connection to the closed circle from the previous piece at $𝒙=−1$
        • At $𝒙=1$
    • $𝒇(𝒙)=𝒙$ for $𝒙\ge 1$
      • Graph of $𝒚=𝒙$ (a straight line through the origin with slope 1)
      • Only consider the portion for $𝒙\ge 1$
      • Closed circle at $𝒙=1$ due to “≥”

Ｄｏｍａｉｎ　＆　Ｒａｎｇｅ

• [35:25]. Domain and Range of a function – [📷image]
  • Definition of Domain
    • All values that can be “input” into a function
    • Typically 𝒙-values, but can vary depending on the problem’s variables
  • Definition of Range
    • All possible values that are “output” by a function
    • Typically 𝒚-values or 𝒇(𝒙)
  • Restrictions on the Domain
    • Real-world constraints (problem context)
      • [36:40]. 🧩 Example – The area of a square (side 𝓼): $𝓼^{2}$
        • The domain (𝓼) cannot be negative
        • The side can be zero (trivial area)
    • [38:06]. Formula-based restrictions
      • Division by zero: the denominator cannot be zero (e.g., $𝒚=\dfrac{1}{𝒙}$)
      • Even-index roots: the radicand cannot be negative in real numbers (e.g., $𝒇(𝒙)=\sqrt{𝒙}$)
        • The square root of zero is defined
        • In complex numbers, roots of negative numbers do exist (using $i$)
  • [39:30]. Natural Domain
    • Definition for this course: all values that work in the formula, including natural and formula-based restrictions
      • When we find the natural domain, we must consider (include) any restrictions that naturally arise from the formula — these restrictions define which values must be excluded.
    • In short: all values for which the formula makes sense
• [40:54]. How to find the Natural Domain
  • 🧩 Example – 1 Natural Domain: $𝒇(𝒙)=𝒙^{3}$ – [📷image]
    • Is there any number you cannot plug in? No
    • Domain: all real numbers ($𝓡$ or $𝒙\in 𝓡$)
  • [42:21]. 🧩 Example – 2 Natural Domain: $𝒇(𝒙)=\dfrac{2𝒙+1}{(𝒙-1)(𝒙-3)}$ – [📷image]
    • Some numbers make the denominator zero; these are excluded
    • Set the denominator $(𝒙-1)(𝒙-3)=0
      • Solutions: $𝒙=1$ and $𝒙=3$
    • Domain:
      • Sets: $𝒙\in 𝓡\setminus\{1,3\}$
      • Set-Builder Notation: $\{\, 𝓍\in 𝓡 : 𝓍\neq \mathbf{1}\ \text{and}\ 𝓍\neq \mathbf{3}\,\}$
      • Union of Intervals: $(-\infty,1)\cup(1,3)\cup(3,\infty)$
  • [45:15]. 🧩 Example – 3 Natural Domain: $𝓰(𝒙)=\tan(𝒙)$ – [📷image]
    • Definition of tangent: $\tan(𝒙)=\dfrac{\sin(𝒙)}{\cos(𝒙)}$
    • Tangent is undefined where cosine is zero
    • Cosine is zero at $𝒙=\dfrac{\pi}{2},\ \dfrac{3\pi}{2},\ -\dfrac{\pi}{2},\ \dots$
    • Domain: all real numbers except $𝒙=\dfrac{\pi}{2}+n\pi$, where $n$ is an integer
  • [48:10]. 🧩 Example – 4 Natural Domain: $𝒇(𝒙)=\sqrt{6-5𝒙+𝒙^{2}}$ – [📷image]
    • Potential problem: the square root of a negative number
    • The radicand must be $\ge 0$: $6-5𝒙+𝒙^{2}\ge 0$
    • Factorization of the quadratic: $(𝒙-2)(𝒙-3)\ge 0$
    • Sign analysis test to find intervals satisfying this inequality
      • Critical points: $𝒙=2$ and $𝒙=3$
      • Draw a number line with critical points
      • Test values in each interval
        • For $𝒙<2$ (e.g., $0$), the product is positive
        • For $2<𝒙<3$ (e.g., $2.5$), the product is negative
        • For $𝒙>3$ (e.g., $4$), the product is positive
      • Intervals where the expression is nonnegative form the domain
        • Domain in interval notation: $(-\infty,2]\cup[3,\infty)$
  • [58:50]. 🧩 Example – 5 Natural Domain: $𝒇(𝒙)=\dfrac{𝒙^{2}-4}{𝒙-2}$ – [📷image]
    • Potential problem: denominator is zero if $𝒙=2$
    • Domain: $𝒙\in 𝓡\setminus\{2\}$
    • Simplification: $𝒇(𝒙)=\dfrac{(𝒙+2)(𝒙-2)}{𝒙-2}=𝒙+2$ (for $𝒙\neq 2$)
    • Even if the function simplifies, the original domain restriction remains
    • The discontinuity at $𝒙=2$ is called a “hole” or removable discontinuity
      • Occurs when the problematic factor can be canceled
      • In polynomials, this happens when substitution yields $0/0$
      • [01:03:14]. Graph
    • If the problematic factor cannot be canceled, we get a vertical asymptote
      • Typically a number/zero form upon substitution
    • [01:06:00]. Removable discontinuity vs. Vertical asymptote – [📷image]
      • $𝒇(𝒙)=\dfrac{𝒙^{2}-4}{𝒙-2}$
        • Has a “domain problem” at $𝒙=2$
        • But it can be removed, problematic factor can be canceled (this is a removable discontinuity — it creates a "hole")
      • $𝓰(𝒙)=\dfrac{3𝒙}{𝒙-4}$
        • Also has a “domain issue” at $𝒙=4$
        • But it cannot be removed, problematic factor cannot be canceled — it's a vertical asymptote
  • [01:10:20]. 🧩 Example – 6: $𝒇(𝒙)=2+\sqrt{𝒙-1}$ – [📷image]
    • Potential problem: negative radicand
    • Inequality: $𝒙-1\ge 0$
    • Solution: $𝒙\ge 1$
    • Domain in interval notation: $[1,\infty)$
  • [01:13:00]. 🧩 Example – 7: with a root in the denominator: $𝒇(𝒙)=\dfrac{1}{\sqrt{𝒙-1}}$ – [📷image]
    • Restrictions: the radicand cannot be negative and the denominator cannot be zero
    • Inequality: $𝒙-1>0$
    • Solution: $𝒙>1$
• [01:13:35]. How to find the Range
  • The set of all possible outputs (𝒚-values)
  • 𝓕𝓲𝓻𝓼𝓽 𝓶𝓮𝓽𝓱𝓸𝓭: substitute domain values into the function and observe the results
    • [01:15:55]. 🧩 Example – : $𝒇(𝒙)=2+\sqrt{𝒙-1}$ with domain $[1,\infty)$
      • If $𝒙=1$, $𝒇(1)=2$
      • As $𝒙$ decreases, $𝒇(𝒙)$ increases toward $+\infty$
      • Range: $[2,\infty)$
  • 𝓢𝓮𝓬𝓸𝓷𝓭 𝓶𝓮𝓽𝓱𝓸𝓭 (not always easy): solve for $𝒙$ in terms of $𝒚$ and find that domain of $𝒚$
    • This makes the outputs act like “inputs”
  • [01:16:15]. 🧩 Example – : $𝒇(𝒙)=\dfrac{𝒙+1}{𝒙-1}$ – [📷image]
    • Domain: $𝒙\neq 1$, vertical asymptote (problematic factor cannot be canceled)
    • Solve for $𝒙$: $𝒚(𝒙-1)=𝒙+1 \;\Rightarrow\; 𝒚𝒙-𝒚=𝒙+1 \;\Rightarrow\; 𝒚𝒙-𝒙=𝒚+1 \;\Rightarrow\; 𝒙(𝒚-1)=𝒚+1 \;\Rightarrow\; 𝒙=\dfrac{𝒚+1}{𝒚-1}$
    • The condition $𝒚\neq 1$ indicates the range excludes $𝒚=1$ (the domain of $𝒚$ is the range of the function)
    • That exclusion ($𝒚\neq 1$) corresponds to a horizontal asymptote
    • Warning: this method is not always straightforward

Ａｐｐｌｉｃａｔｉｏｎ

• [01:21:25] 🧩 Example – Application problem: Cardboard box – [📷image]
  • Problem description: cutting squares out of the corners of a piece of cardboard to form a box
  • Cardboard dimensions: 16 inches by 30 inches
  • The cut squares must have the same side length ($𝒙$) on each corner
  • Formula for the volume of the box as a function of $𝒙$
    • Length of the base: $30-2𝒙$
    • Width of the base: $16-2𝒙$
    • Height of the box: $𝒙$
    • Volume $𝓥(𝒙)=𝒙(16-2𝒙)(30-2𝒙)$
  • [01:27:05] Realistic domain for the volume function
    • There are no denominators or roots in the volume formula
    • Real-world restriction: the length of the cuts ($𝒙$) must be positive ($𝒙>0$)
    • The cut cannot be so large that a side is fully removed
    • The maximum cut is less than half of the shorter dimension (16 inches), so $𝒙<8$< /li>
    • Realistic domain: $0<𝒙<8$< /li>

Ｅｖｅｎ　ａｎｄ　Ｏｄｄ　ｆｕｎｃｔｉｏｎｓ

• [01:30:50] Brief review – [📷image]
  • Even function
    • Algebraic definition: $𝒇(-𝒙)=𝒇(𝒙)$
    • Symmetry about the y-axis
    • Example: $𝒇(𝒙)=𝒙^{4}-𝒙^{2}+1$
  • Odd function
    • Algebraic definition: $𝓰(-𝒙)=-𝓰(𝒙)$
    • Symmetry about the origin (180° rotation)
    • Example: $𝓰(𝒙)=𝒙^{3}-𝒙$
  • Method to check if a function is even or odd: substitute $-𝒙$ and simplify