Concept of a Function: Domain & Range

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✖️ 1. Definition of a function as a mapping (input/output) and the vertical line test

🎯 What Makes a Function

A function pairs each input with exactly one output.
Think of it like a vending machine: one button press gives one specific snack.
If an input could give two different outputs, it's NOT a function.
Vertical line test: Draw a vertical line anywhere on the graph.
If the line touches the graph more than once, it fails (not a function).

Example: $x = 4$ gives $y = 2$ (function), but $x = 4$ giving both $y = 2$ and $y = -2$ is NOT a function.

💡 One input → One output only!

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1. Definition of a function as a mapping (input/output) and the vertical line test

Function as a Mapping

A function is a rule that assigns to each input exactly one output. We write $f: A \to B$ to indicate that $f$ maps elements from set $A$ (inputs) to set $B$ (outputs).

Intuition: Think of a function as a machine: you feed it one value, and it returns exactly one result. No input can produce two different outputs.

Core Rules:

Each input must correspond to exactly one output.
Multiple inputs may map to the same output (many-to-one is allowed).
If any vertical line intersects the graph at more than one point, the relation is not a function (vertical line test).

Consequence: The vertical line test provides a quick visual check for whether a graph represents a function.

Example: Consider $f(x) = 2x + 1$ . For input $x = 3$ , the output is $f(3) = 7$ . Each $x$ produces exactly one $f(x)$ , so this is a function.

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Based on the definition of a function as a mapping, which statement correctly describes how inputs and outputs must relate?

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✖️ 2. Function notation: evaluating $f(x)$ for numerical and algebraic inputs

🔢 Reading Function Notation

$f(x)$ means "the output when you plug in $x$ ".
To evaluate $f(3)$ , replace every $x$ in the formula with $3$ .
For $f(x) = 2x + 1$ , we get $f(3) = 2(3) + 1 = 7$ .
You can also plug in expressions: $f(a + 1)$ means replace $x$ with $a + 1$ .
Always use parentheses when substituting to avoid mistakes.

Example: If $f(x) = x^2 - 4$ , then $f(5) = 5^2 - 4 = 25 - 4 = 21$ .

💡 Replace $x$ everywhere, then calculate!

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2. Function notation: evaluating $f(x)$ for numerical and algebraic inputs

Function Notation and Evaluation

The notation $f(x)$ represents the output of function $f$ when the input is $x$ . To evaluate $f(a)$ , substitute $a$ for every occurrence of $x$ in the function's formula.

Intuition: The symbol $f(x)$ is not multiplication; it means "the value of $f$ at $x$ ."

Core Rules:

Replace the variable with the given input value (numerical or algebraic).
Simplify using order of operations.
For algebraic inputs like $f(x + 2)$ , substitute the entire expression $(x + 2)$ wherever $x$ appears.

Consequence: Proper substitution is essential for evaluating compositions and transformations of functions.

Example: If $f(x) = x^2 - 3x$ , then $f(4) = 4^2 - 3(4) = 16 - 12 = 4$ . For algebraic input, $f(a + 1) = (a + 1)^2 - 3(a + 1) = a^2 - a - 2$ .

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Given the function $f(x) = x^2 - 5x$ , evaluate $f(6)$ .

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✖️ 3. Determining Domain (legal inputs) by checking for division by zero or negative roots

⚠️ Finding Legal Inputs (Domain)

Domain = all $x$ values you're allowed to use.
Rule 1: Never divide by zero (bottom of fraction cannot be zero).
Rule 2: Cannot take square root of a negative number (for real numbers).
Set denominators not equal to zero and solve.
Set expressions under square roots greater than or equal to zero.

Example: For $f(x) = \frac{1}{x - 3}$ , domain is all real numbers except $x = 3$ (since $3 - 3 = 0$ ).

💡 Ban the zeros below and negatives under roots!

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3. Determining Domain (legal inputs) by checking for division by zero or negative roots

Domain of a Function

The domain of a function is the set of all input values for which the function produces a valid output. We must exclude values that cause mathematical errors.

Intuition: The domain answers "Which inputs are allowed?" by identifying restrictions imposed by the function's formula.

Core Rules:

Division: Exclude values that make any denominator equal to zero.
Even roots: Exclude values that make the expression under an even root (like $\sqrt{x}$ ) negative.
Logarithms: Exclude non-positive arguments (domain requires argument greater than zero).
If no restrictions exist, the domain is all real numbers.

Consequence: Identifying domain restrictions prevents undefined operations and ensures the function is well-defined.

Example: For $f(x) = \frac{1}{x - 5}$ , set $x - 5 \neq 0$ , so $x \neq 5$ . Domain: all real numbers except $5$ .

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Consider the function $f(x) = 1 / (x - 7)$ .

What single numeric value of $x$ must be excluded from the domain to prevent division by zero?

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✖️ 4. Understanding Range (possible outputs) through verbal and graphical analysis

📊 Finding Possible Outputs (Range)

Range = all possible $y$ values the function can produce.
Look at the graph: what heights does it reach?
For $y = x^2$ , outputs are never negative (range is $y \geq 0$ ).
Horizontal lines on the graph show which $y$ values are hit.
Some functions have restricted ranges even with unlimited domains.

Example: $f(x) = x^2 + 1$ has range $y \geq 1$ because squaring gives $0$ or more, then we add $1$ .

💡 Range = vertical span of the graph!

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4. Understanding Range (possible outputs) through verbal and graphical analysis

Range of a Function

The range of a function is the set of all possible output values that the function can produce as the input varies over the domain.

Intuition: While domain asks "What can go in?", range asks "What can come out?" by examining all achievable $f(x)$ values.

Core Rules:

Analyze the function's behavior: Does it have a minimum or maximum value?
Use graphical analysis: the range corresponds to the vertical extent of the graph.
For algebraic functions, solve $y = f(x)$ for $x$ and identify restrictions on $y$ .
Transformations (shifts, stretches) directly affect the range.

Consequence: Understanding range is crucial for modeling real-world constraints where outputs have physical limits.

Example: For $f(x) = x^2$ , since squares are never negative, the range is $[0, \infty)$ . Graphically, the parabola extends upward from $y = 0$ .

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Consider the function $f(x) = x^2 + 5$ . What is the minimum possible output value of this function?

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✖️ 5. Applications: Defining operational constraints for machines (e.g., maximum load as a function of input power)

🏭 Real-World Function Limits

Machines have physical constraints that define domain and range.
A crane's max load might be $L(p) = 50p$ where $p$ is power in kilowatts.
Domain constraint: power cannot be negative ( $p \geq 0$ ).
Range constraint: crane cannot lift more than 5000 kilograms (so $L \leq 5000$ ).
Always check manufacturer limits to define realistic domains.

Example: If max power is 80 kilowatts, domain is $0 \leq p \leq 80$ and range is $0 \leq L \leq 4000$ kilograms.

💡 Real machines = real boundaries on inputs and outputs!

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5. Applications: Defining operational constraints for machines (e.g., maximum load as a function of input power)

Operational Constraints in Applied Functions

In real-world applications, functions model relationships between physical quantities, and domain/range represent operational constraints—the safe or feasible limits of a system.

Intuition: A machine's specifications define which inputs are allowed (domain) and what outputs are achievable (range).

Core Rules:

Domain constraints arise from physical limitations: power cannot be negative, time cannot be negative, capacity has upper bounds.
Range constraints reflect output limits: a motor's maximum load, a tank's volume capacity.
Always interpret domain and range in context of the problem's units and meaning.

Consequence: Violating domain constraints can lead to system failure; understanding range ensures realistic expectations.

Example: A crane's load capacity $L(p) = 50p$ (in kg) depends on power $p$ (in kW). If the crane operates between 2 kW and 10 kW, domain is $[2, 10]$ and range is $[100, 500]$ kg.

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A water pump's output volume is modeled by a function $V(t)$ , where $t$ is the operating time in hours. The pump cannot run for more than $8$ hours before it overheats and shuts down. Which interval represents the realistic domain for this function?

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Concept of a function: domain and range

✖️ 1. Definition of a function as a mapping (input/output) and the vertical line test

🎯 What Makes a Function

1. Definition of a function as a mapping (input/output) and the vertical line test

Function as a Mapping

✖️ 2. Function notation: evaluating $f(x)$ for numerical and algebraic inputs

🔢 Reading Function Notation

2. Function notation: evaluating $f(x)$ for numerical and algebraic inputs

Function Notation and Evaluation

✖️ 3. Determining Domain (legal inputs) by checking for division by zero or negative roots

⚠️ Finding Legal Inputs (Domain)

3. Determining Domain (legal inputs) by checking for division by zero or negative roots

Domain of a Function

✖️ 4. Understanding Range (possible outputs) through verbal and graphical analysis

📊 Finding Possible Outputs (Range)

4. Understanding Range (possible outputs) through verbal and graphical analysis

Range of a Function

✖️ 5. Applications: Defining operational constraints for machines (e.g., maximum load as a function of input power)

🏭 Real-World Function Limits

5. Applications: Defining operational constraints for machines (e.g., maximum load as a function of input power)

Operational Constraints in Applied Functions

AWAITING_CONFIRMATION

✖️ 1. Definition of a function as a mapping (input/output) and the vertical line test

🎯 What Makes a Function

1. Definition of a function as a mapping (input/output) and the vertical line test

Function as a Mapping

✖️ 2. Function notation: evaluating f(x)f(x)f(x) for numerical and algebraic inputs

🔢 Reading Function Notation

2. Function notation: evaluating f(x)f(x)f(x) for numerical and algebraic inputs

Function Notation and Evaluation

✖️ 3. Determining Domain (legal inputs) by checking for division by zero or negative roots

⚠️ Finding Legal Inputs (Domain)

3. Determining Domain (legal inputs) by checking for division by zero or negative roots

Domain of a Function

✖️ 4. Understanding Range (possible outputs) through verbal and graphical analysis

📊 Finding Possible Outputs (Range)

4. Understanding Range (possible outputs) through verbal and graphical analysis

Range of a Function

✖️ 5. Applications: Defining operational constraints for machines (e.g., maximum load as a function of input power)

🏭 Real-World Function Limits

5. Applications: Defining operational constraints for machines (e.g., maximum load as a function of input power)

Operational Constraints in Applied Functions

AWAITING_CONFIRMATION

✖️ 2. Function notation: evaluating $f(x)$ for numerical and algebraic inputs

2. Function notation: evaluating $f(x)$ for numerical and algebraic inputs