Solved Exercises on What is a Function in Integrated Math 1

Master functions: definition, domain, range, function notation, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Function identification from ordered pairs
Exercise 1
Determine whether the relation R = {(1,2), (3,4), (5,6), (1,8)} is a function. Explain your reasoning.
Definition:

Function: A relation where each input (domain element) is paired with exactly one output (range element).

Ordered pair: A pair (x,y) where x is the input and y is the output.

Function test method:
  1. List all ordered pairs
  2. Identify the input values (x-coordinates)
  3. Check if any input appears more than once
  4. If yes, it's not a function; if no, it is a function
Relation
R = {(1,2), (3,4), (5,6), (1,8)}
Inputs
{1, 3, 5, 1}
Result
Not a function
Step 1: List all ordered pairs

R = {(1,2), (3,4), (5,6), (1,8)}

Step 2: Identify input values

The input values (x-coordinates) are: 1, 3, 5, 1

Step 3: Check for repeated inputs

The input 1 appears twice: once paired with 2 and once paired with 8

Step 4: Apply function definition

Since input 1 maps to two different outputs, this violates the function definition

R is NOT a function
Final answer:

The relation R = {(1,2), (3,4), (5,6), (1,8)} is NOT a function.

Explanation: The input 1 is paired with two different outputs (2 and 8), violating the function rule that each input must have exactly one output.

Applied rules:

• Function definition: Each input → exactly one output

• Vertical line test: If any vertical line intersects the graph more than once, it's not a function

• One-to-one mapping: Each domain element maps to only one range element

2 Function notation and evaluation
Exercise 2
Given f(x) = 2x + 5, find f(3), f(-1), and f(a + 1). Then state the domain and range of f(x).
Definition:

Function notation: f(x) reads as "f of x" and represents the output when x is the input.

Domain: The set of all possible input values.

Range: The set of all possible output values.

Function
f(x) = 2x + 5
Evaluations
f(3)=11, f(-1)=3, f(a+1)=2a+7
Domain/Range
Domain: ℝ, Range: ℝ
Step 1: Evaluate f(3)

f(3) = 2(3) + 5 = 6 + 5 = 11

Step 2: Evaluate f(-1)

f(-1) = 2(-1) + 5 = -2 + 5 = 3

Step 3: Evaluate f(a + 1)

f(a + 1) = 2(a + 1) + 5 = 2a + 2 + 5 = 2a + 7

Step 4: Determine domain and range

Domain: All real numbers (ℝ) since there are no restrictions

Range: All real numbers (ℝ) since it's a linear function with non-zero slope

f(3) = 11, f(-1) = 3, f(a+1) = 2a + 7
Final answer:

f(3) = 11, f(-1) = 3, f(a+1) = 2a + 7

Domain: All real numbers (ℝ)

Range: All real numbers (ℝ)

Applied rules:

• Function evaluation: Replace x with the given value

• Domain: Values for which the function is defined

• Range: Possible output values

3 Function from a table
Exercise 3
x-2013
y4127
Is this table representing a function? If so, identify the domain and range.
Definition:

Function table: A table where each input x corresponds to exactly one output y.

Domain: Set of all x-values in the table.

Range: Set of all y-values in the table.

Table
x: {-2, 0, 1, 3}, y: {4, 1, 2, 7}
Analysis
Each x has unique y
Result
Yes, it's a function
Step 1: List input values (x)

Input values: -2, 0, 1, 3

Step 2: Check for repeated inputs

No input value appears more than once

Step 3: Verify each input has exactly one output

-2 → 4, 0 → 1, 1 → 2, 3 → 7

Step 4: Identify domain and range

Domain = {-2, 0, 1, 3}, Range = {1, 2, 4, 7}

Yes, it's a function
Final answer:

Yes, this table represents a function.

Domain = {-2, 0, 1, 3}

Range = {1, 2, 4, 7}

Applied rules:

• Function test: Each input must have exactly one output

• Domain: All x-values in the table

• Range: All y-values in the table

What is a Function? Complete Guide
\(f: X \rightarrow Y\)
Function Notation
Key definitions:

Function: A relation where each input is paired with exactly one output

Domain: Set of all possible input values (x-values)

Range: Set of all possible output values (y-values)

Function notation: f(x) represents the output when x is the input

Complete methodology:
  1. Identify inputs and outputs: Determine what values go in and come out
  2. Apply function test: Check if each input has exactly one output
  3. Find domain and range: Identify all possible input and output values
  4. Evaluate function: Substitute values to find specific outputs
Tip 1: Think of a function as a machine that takes an input and produces a unique output
Tip 2: Use the vertical line test on graphs: if any vertical line touches the graph more than once, it's not a function
Tip 3: In ordered pairs, check that no x-value repeats with different y-values
Tip 4: Domain restrictions occur when denominators equal zero or when taking even roots of negative numbers
Common errors: Confusing domain and range, thinking functions must be linear, not checking for repeated inputs
Exam preparation: Practice identifying functions from various representations (tables, graphs, ordered pairs, equations)
Formulas to know by heart:

• Function definition: Each input → exactly one output

• Domain: Set of all possible x-values

• Range: Set of all possible y-values

• Function notation: f(x) = output when x is input

Solution: Exercises 4 to 5
4 Real-world function
Exercise 4
The cost C (in dollars) of renting a car is given by C(t) = 50t + 20, where t is the number of days. Is this a function? Find C(3) and interpret its meaning. What are the domain and range in the context of this problem?
Definition:

Real-world function: A function that models a practical situation with meaningful inputs and outputs.

Contextual domain: Domain restricted by practical considerations.

Function
C(t) = 50t + 20
Evaluation
C(3) = 170
Domain/Range
Domain: t ≥ 0, Range: C ≥ 20
Step 1: Verify it's a function

For every input t, there is exactly one output C(t) = 50t + 20

Step 2: Evaluate C(3)

C(3) = 50(3) + 20 = 150 + 20 = 170

Step 3: Interpret C(3) = 170

It costs $170 to rent the car for 3 days

Step 4: Determine contextual domain and range

Domain: t ≥ 0 (can't rent for negative days)

Range: C ≥ 20 (minimum cost is $20, the base fee)

C(3) = $170
Final answer:

Yes, C(t) = 50t + 20 is a function.

C(3) = $170, which means it costs $170 to rent the car for 3 days.

Domain: t ≥ 0 (days), Range: C ≥ 20 (dollars)

Applied rules:

• Real-world constraints: Domain and range may be limited by context

• Function evaluation: Substitution gives specific output

• Interpretation: Connect mathematical results to real-world meaning

5 Function from a mapping diagram
Exercise 5
Consider the mapping: A = {1, 2, 3, 4} → B = {a, b, c, d} where 1→a, 2→b, 3→c, 4→a. Is this a function? Explain. What is the domain and range?
Definition:

Mapping diagram: A visual representation showing how elements from one set map to another set.

Function mapping: Each element in the domain maps to exactly one element in the range.

Mapping
1→a, 2→b, 3→c, 4→a
Analysis
Each input maps once
Result
Yes, it's a function
Step 1: List all mappings

1 → a, 2 → b, 3 → c, 4 → a

Step 2: Check each input

1 maps to a (once), 2 maps to b (once), 3 maps to c (once), 4 maps to a (once)

Step 3: Apply function definition

Each input in A maps to exactly one output in B

Step 4: Identify domain and range

Domain = {1, 2, 3, 4}, Range = {a, b, c} (note: a appears twice as output, but is counted once in range)

Yes, it's a function
Final answer:

Yes, this is a function.

Domain = {1, 2, 3, 4}

Range = {a, b, c}

Note: Although 'a' appears as output for both 1 and 4, the range contains each distinct output value only once.

Applied rules:

• Mapping test: Each domain element maps to exactly one range element

• Range uniqueness: Only distinct output values are listed

• Function verification: Check that no input maps to multiple outputs

Function Fundamentals: Complete Summary
\(f: X \rightarrow Y, \text{ where } x \mapsto f(x)\)
Function Mapping
Key definitions:

Function: A relation where each input (domain element) is paired with exactly one output (range element)

Domain: The set of all possible input values

Range: The set of all possible output values

Function notation: f(x) represents the output when x is the input

Independent variable: Input variable (usually x)

Dependent variable: Output variable (usually y or f(x))

Complete methodology:
  1. Identify inputs and outputs: Determine what values go in and come out
  2. Apply function test: Check if each input has exactly one output
  3. Find domain and range: Identify all possible input and output values
  4. Evaluate function: Substitute values to find specific outputs
  5. Interpret results: Connect mathematical results to real-world meaning
Tip 1: Think of a function as a machine that takes an input and produces a unique output
Tip 2: Use the vertical line test on graphs: if any vertical line touches the graph more than once, it's not a function
Tip 3: In ordered pairs, check that no x-value repeats with different y-values
Tip 4: Domain restrictions occur when denominators equal zero or when taking even roots of negative numbers
Tip 5: Remember that different inputs can have the same output (many-to-one is allowed)
Common errors: Confusing domain and range, thinking functions must be linear, not checking for repeated inputs
Exam preparation: Practice identifying functions from various representations (tables, graphs, ordered pairs, equations)
Formulas to know by heart:

• Function definition: Each input → exactly one output

• Domain: Set of all possible x-values

• Range: Set of all possible y-values

• Function notation: f(x) = output when x is input

Function Representations Comparison
Function Analysis: Different Representations
Consider the function f(x) = x² - 2x + 1:
• Algebraic: f(x) = x² - 2x + 1
• Tabular: x values: -1, 0, 1, 2, 3; corresponding f(x) values
• Graphical: Parabolic curve
All representations describe the same function.

Analysis: The same function can be represented in multiple ways, each providing different insights.

  • Algebraic form allows precise calculations
  • Tabular form shows specific input-output pairs
  • Graphical form reveals overall behavior and trends

Questions & Answers

Question: I understand that each input must have exactly one output, but can different inputs have the same output? Like in f(x) = x², both x = 2 and x = -2 give f(x) = 4.

Answer: Yes, absolutely! This is a common source of confusion. The function definition requires:

  • One-to-one restriction: Each input → exactly one output (this is mandatory)
  • Many-to-one allowed: Multiple inputs → same output (this is allowed)

In your example f(x) = x²:

  • Input 2 → Output 4 (allowed)
  • Input -2 → Output 4 (also allowed)
  • Input 2 → Both 4 AND some other value (not allowed)

This is why the vertical line test works: it checks if any x-value maps to multiple y-values, not if multiple x-values map to the same y-value.

Question: How do I find the domain of a function if it's not explicitly stated? For example, what's the domain of f(x) = 1/(x-3)?

Answer: To find the domain, identify values that would make the function undefined:

  • Division by zero: Denominator cannot equal zero
  • Even roots of negatives: Cannot take √(negative number)
  • Logarithms of non-positive numbers: log(x) requires x > 0

For f(x) = 1/(x-3):

  • Set denominator equal to zero: x - 3 = 0
  • Solve: x = 3
  • Exclude this value from the domain
  • Domain: All real numbers except x = 3, written as ℝ \ {3} or x ≠ 3

For polynomial functions like f(x) = x² + 3x + 2, the domain is all real numbers since there are no restrictions.

Question: How can I quickly determine if a graph represents a function?

Answer: Use the Vertical Line Test:

  1. Imagine moving a vertical line across the entire graph
  2. If the vertical line ever intersects the graph at more than one point, it's NOT a function
  3. If the vertical line intersects the graph at most once at any position, it IS a function

Examples:

  • Parabolas opening up or down: Function (vertical line hits once or twice at vertex)
  • Circles: Not a function (vertical line can hit twice)
  • Lines: Function (vertical line hits once, unless the line is vertical)

Remember: This test verifies that each x-value corresponds to at most one y-value.