Study Guide on Functions and Linear Notations

Definition of Symbols: A function is commonly denoted by symbols such as $f$ , $g$ , or other letters.
Mathematical Notation: A function $f$ from set $X$ to set $Y$ is written as $f: X \rightarrow Y$ . This is read as " $f$ is a function from $X$ to $Y$ ."
Structural Components:
- Domain: Represented by the set $X$ , containing all input values.
- Codomain: Represented by the set $Y$ , containing all potential output values.
- Function Rule: The specific mathematical operation applied to the input, such as $f(x) = 2x$ .
The Image: If an ordered pair $(x, y)$ belongs to the function $f$ , then the element $y$ is called the image of the element $x$ under the function $f$ .
The Equation: This relationship is expressed as $y = f(x)$ .
- Independent Variable: $X$ is the independent variable because it is the input value chosen from the domain.
- Dependent Variable: $y$ is the dependent variable because its value depends on the chosen value of $X$ .
The Machine Analogy: A function can be viewed as a machine where:
- Inputs: Domain values ( $X$ ) are fed into the machine.
- Process: The function rule ( $f(x)$ ) acts on the input.
- Outputs: Range values ( $y$ ) are produced.
Crucial Distinction: The notation $f(x)$ does not signify $f$ multiplied by $x$ . Instead, it explicitly means "the value of the function $f$ at the point $x$ ." For example, $f(1)$ represents the specific value of the function when $X = 1$ .

Example 1: Basic Mapping:
- Given: $X = \{0, 1, 4\}$ , $Y = \{0, 1, 2, 4\}$ , and the rule $f(x) = 2x$ .
- Calculations:
  - At $X = 0$ , then $f(0) = 2 \times 0 = 0$ .
  - At $X = 1$ , then $f(1) = 2 \times 1 = 2$ .
  - At $X = 4$ , then $f(4) = 2 \times 4 = 8$ .
- Conclusion: The range of the function is $\{0, 2, 8\}$ .
Example 2: Quadratic Rule:
- Given: $X = \{-1, 0, 1\}$ , $Y = \{0, -1, -2\}$ , and the rule $f(x) = x^2 - 1$ .
- Calculations:
  - At $X = -1$ , then $f(-1) = (-1)^2 - 1 = 0$ .
  - At $X = 0$ , then $f(0) = (0)^2 - 1 = -1$ .
  - At $X = 1$ , then $f(1) = (1)^2 - 1 = 0$ .
- Results: The images of the elements are $0$ and $-1$ . The range of the function is $\{0, -1\}$ .
Special Case: Function on a Set:
- If a function maps from set $X$ to itself ( $f: X \rightarrow X$ ), it is stated as " $f$ is a function on $X$ ."
Example 3: Natural Number Function:
- Given: $f: \mathbb{N} \rightarrow \mathbb{N}$ where $\mathbb{N}$ is the set of natural numbers, and $f(x) = x + 1$ .
- Rule: The image of any natural number is the number plus $1$ .
- Points:
  - $f(0) = 0 + 1 = 1$
  - $f(1) = 1 + 1 = 2$
  - $f(2) = 2 + 1 = 3$
  - $f(3) = 3 + 1 = 4$
- Ordered Pairs: $(0, 1), (1, 2), (2, 3), (3, 4)$ .
- Range: The range is all natural numbers except zero, expressed as $\mathbb{N} - \{0\}$ . This is because no natural number, when added to $1$ , results in an output of $0$ .

Definition: A function $f$ defined by $f(x) = ax + b$ , where $a$ and $b$ are real numbers and $a \neq 0$ .
Degree: It is a first-degree function because the highest exponent of the variable $x$ is $1$ .
Domain: In general, the domain is the set of real numbers $\mathbb{R}$ . In real-life applications, it may be a subset of $\mathbb{R}$ based on practical data constraints.
Graphical Representation: Represents as a straight line.
- Y-axis Interaction: Intersects the y-axis at the point $(0, b)$ .
- X-axis Interaction: Intersects the x-axis at the point $(-\frac{b}{a}, 0)$ .
Example Case: For $f(x) = 2x - 5$ :
- Intersection with y-axis: $(0, -5)$ .
- Intersection with x-axis: $(\frac{5}{2}, 0)$ .
Graphing Procedure:
- Find at least two ordered pairs belonging to the function.
- It is recommended to find a third ordered pair to verify that all three points align on a single straight line.
Special Case ( $b = 0$ ):
- The function $f(x) = ax$ is a straight line passing through the origin point $(0, 0)$ .
- Note on Coefficients: If the coefficient of $x$ is a fraction ( $\frac{1}{d}$ ), select values for $x$ that are divisible by the denominator ( $d$ ) to simplify calculations and graphing.

Factory Production Cost:
- Function: $f(x) = 60x + 40$ , where $x$ is the number of shirts and $f(x)$ is total cost in pounds.
- Fixed Cost ( $x=0$ ): $40\,\text{pounds}$ . This is the cost regardless of production volume.
- Variable Cost: $60\,\text{pounds}$ per shirt.
- Finding Production Quantity: If total cost is $940\,\text{pounds}$ , set $60x + 40 = 940$ .
  - $60x = 900$
  - $x = 15\,\text{shirts}$ .
- Domain Constraint: Since you cannot produce negative or partial shirts, the domain is the set of natural numbers $\mathbb{N}$ , and the graph consists of discrete points.
Service Costs (Meal Delivery):
- Example Rule: $f(x) = 50x + 15$ .
- Context: $50\,\text{pounds}$ per meal plus a $15\,\text{pounds}$ fixed delivery fee.
- Logical Constraint: The domain starts at counting numbers (min. 1 meal), as paying a delivery fee for zero meals is illogical.
Battery Depletion Model:
- Function: $f(x) = 100 - 10x$ , where $x$ is hours of use and $f(x)$ is the percentage of remaining charge.
- Domain: Restricted to $[0, 10]$ because battery cannot have negative charge or exceed $100\%$ .
- Finding Usage Time: If remaining charge is $20\%$ , set $20 = 100 - 10x$ .
  - $10x = 80$
  - $x = 8\,\text{hours}$ .

Definition: A function $f$ where $f(x) = b$ , and $b$ is a constant real number.
Degree: If $b \neq 0$ , the function is of zero degree.
Graphical Representation: A straight line parallel to the X-axis.
- Case b > 0: The line is above the X-axis and passes through $(0, b)$ .
- Case b < 0: The line is below the X-axis and passes through $(0, b)$ .
- Case $b = 0$ : The line coincides with the X-axis (the line $y = 0$ ) and passes through $(0, 0)$ .
Examples:
- $f(x) = 2$ : Line passes through $(0, 2)$ and is parallel to the X-axis.
- $f(x) = -2$ : Line passes through $(0, -2)$ and is parallel to the X-axis.
- $f(x) = 0$ : Overlays exactly on the X-axis.

Try it Yourself 1:
- Problem: If $X = \{2, 4, 6, 8\}$ , $Y = \{1, 2, 3, 4, 5, 6\}$ and the function $f: X \rightarrow Y$ where $f(x) = \frac{1}{2}x$ . Write $f$ as a set of ordered pairs, represent it graphically, and find the range.
Try it Yourself 2:
- Problem: If $f: \mathbb{Z} \rightarrow \mathbb{Z}$ (integers), and $f(x) = x^2 - 2$ . Find $f(0), f(1), f(2), f(3), f(4)$ and represent them on a Cartesian grid $\mathbb{Z} \times \mathbb{Z}$ .
Try it Yourself 3:
- Problem: If the line representing $f(x) = 4x - a$ cuts the X-axis at the point $(2, b)$ , find the value of $a$ and $b$ .
Mathematical Problem Solving (Example):
- Question: If the line representing $f(x) = ax + b$ intersects the y-axis at $(0, 3)$ and $f(2) = 7$ , find the values of $a$ and $b$ .
- Solution Step 1: Intersecting the y-axis at $(0, 3)$ means when $x = 0$ , $y = 3$ . Substituting into $f(x) = ax + b$ : $3 = a(0) + b$ , therefore $b = 3$ .
- Solution Step 2: Use $f(x) = ax + 3$ . Since $f(2) = 7$ , substitute $2$ for $x$ and $7$ for $f(x)$ : $7 = 2a + 3$ .
- Solution Step 3: Solve for $a$ : $2a = 4$ , so $a = 2$ .