Solved Exercises on Inputs and Outputs of Functions

Master inputs and outputs of functions: domain, range, function evaluation, and input-output relationships through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Function Evaluation and Input-Output Tables
Exercise 1
Given f(x) = 3x - 2, complete the input-output table and identify the domain and range for the given inputs: x ∈ {-2, 0, 1, 3, 5}.
Definition:

Input: The x-value (domain element) fed into the function

Output: The f(x)-value (range element) produced by the function

Domain: Set of all possible input values

Range: Set of all possible output values

Method for input-output tables:
  1. List all input values in the first column
  2. Apply the function rule to each input
  3. Record the corresponding output in the second column
  4. Identify the domain and range from the table
Function
f(x) = 3x - 2
Process
Multiply by 3, subtract 2
Step 1: Evaluate for each input

f(-2) = 3(-2) - 2 = -6 - 2 = -8

f(0) = 3(0) - 2 = 0 - 2 = -2

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

f(3) = 3(3) - 2 = 9 - 2 = 7

f(5) = 3(5) - 2 = 15 - 2 = 13

Step 2: Create the input-output table
Input (x) Output f(x)
-2 -8
0 -2
1 1
3 7
5 13
Step 3: Identify domain and range

Domain: {-2, 0, 1, 3, 5} (given inputs)

Range: {-8, -2, 1, 7, 13} (corresponding outputs)

Step 4: Analyze the relationship

For the function f(x) = 3x - 2:

Domain (if unrestricted): All real numbers

Range (if unrestricted): All real numbers

But for our specific inputs: Domain = {-2, 0, 1, 3, 5}, Range = {-8, -2, 1, 7, 13}

Domain: {-2, 0, 1, 3, 5}
Range: {-8, -2, 1, 7, 13}
Final answer:

Input-Output Table:

Input (x) Output f(x)
-2 -8
0 -2
1 1
3 7
5 13

Domain: {-2, 0, 1, 3, 5}

Range: {-8, -2, 1, 7, 13}

Applied rules:

• Function evaluation: Substitute each input value into the function rule

• Domain identification: Collect all input values

• Range identification: Collect all corresponding output values

2 Finding Inputs from Outputs
Exercise 2
Given g(x) = x² - 4, find all x-values for which g(x) = 5. Also find g(√5) and g(-√5).
Definition:

Reverse mapping: Finding inputs when outputs are known

Solving equations: Set function equal to desired output and solve for x

Function
g(x) = x² - 4
Equation
x² - 4 = 5
Solution
x = ±3
Step 1: Find x-values when g(x) = 5

Set g(x) = 5: x² - 4 = 5

Add 4 to both sides: x² = 9

Take square root: x = ±√9 = ±3

So x = 3 or x = -3

Step 2: Verify solutions

g(3) = 3² - 4 = 9 - 4 = 5 ✓

g(-3) = (-3)² - 4 = 9 - 4 = 5 ✓

Step 3: Find g(√5)

g(√5) = (√5)² - 4 = 5 - 4 = 1

Step 4: Find g(-√5)

g(-√5) = (-√5)² - 4 = 5 - 4 = 1

Step 5: Note the relationship

Both √5 and -√5 produce the same output (1) because squaring eliminates the sign

This shows that g is not one-to-one

x = ±3 when g(x) = 5
g(√5) = 1
g(-√5) = 1
Final answer:

g(x) = 5 when x = 3 or x = -3

g(√5) = 1

g(-√5) = 1

Applied rules:

• Reverse evaluation: Set function equal to target output and solve

• Square root property: x² = a → x = ±√a

• Even functions: f(-x) = f(x), so opposite inputs may yield same output

3 Domain and Range from Function Rules
Exercise 3
Find the domain and range of h(x) = √(x + 2). Then find h(2), h(7), and h(-2).
Definition:

Domain restriction: Values of x for which the function is defined

Range restriction: Possible output values based on the function rule

Function
h(x) = √(x + 2)
Domain
x ≥ -2
Range
y ≥ 0
Step 1: Find the domain

For h(x) = √(x + 2), the expression under the square root must be non-negative

So: x + 2 ≥ 0

Therefore: x ≥ -2

Domain: [-2, ∞)

Step 2: Find the range

Since √(x + 2) ≥ 0 for all x in the domain

The smallest value occurs when x = -2: h(-2) = √(0) = 0

As x increases, h(x) increases without bound

Range: [0, ∞)

Step 3: Evaluate h(2)

h(2) = √(2 + 2) = √4 = 2

Step 4: Evaluate h(7)

h(7) = √(7 + 2) = √9 = 3

Step 5: Evaluate h(-2)

h(-2) = √(-2 + 2) = √0 = 0

Step 6: Verify domain restrictions

h(2) is valid since 2 ≥ -2 ✓

h(7) is valid since 7 ≥ -2 ✓

h(-2) is valid since -2 ≥ -2 ✓

h(-3) would be invalid since -3 < -2 (negative under square root)

Domain: [-2, ∞)
Range: [0, ∞)
h(2) = 2
h(7) = 3
h(-2) = 0
Final answer:

Domain: x ≥ -2 (or [-2, ∞))

Range: y ≥ 0 (or [0, ∞))

h(2) = 2

h(7) = 3

h(-2) = 0

Applied rules:

• Square root domain: Expression under √ must be ≥ 0

• Range analysis: Consider minimum and maximum possible outputs

• Function behavior: Analyze how function changes over its domain

Inputs and Outputs of Functions: Complete Guide
f: X → Y
Function Mapping
Input
x
Independent variable
Output
f(x)
Dependent variable
Domain
X
Set of inputs
Key definitions:

Function: A relation where each input has exactly one output

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

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

One-to-one function: Each output corresponds to exactly one input

Input-output analysis methodology:
  1. Identify the function rule: Determine f(x) expression
  2. Find domain restrictions: Identify values where function is undefined
  3. Evaluate for inputs: Substitute x-values into function
  4. Find outputs for inputs: Calculate corresponding f(x) values
  5. Solve reverse problems: Find inputs when outputs are known
  6. Determine range: Analyze possible output values
Tip 1: Always check domain restrictions before evaluating functions.
Tip 2: For inverse problems (finding inputs), set function equal to output and solve.
Tip 3: Different functions have different domain restrictions (radicals, fractions, etc.).
Tip 4: Graph the function to visualize input-output relationships.
Common errors: Forgetting domain restrictions, making calculation mistakes, confusing domain and range.
Key insights: Domain affects range, some functions aren't one-to-one, restrictions depend on function type.
Essential formulas and rules:

• Domain restrictions: √(expression) → expression ≥ 0, 1/(expression) → expression ≠ 0

• Range determination: Analyze function behavior and critical points

• Reverse evaluation: If f(x) = y, solve for x

• Function behavior: Linear → all reals, Quadratic → depends on vertex, Radical → restricted

Solution: Exercises 4 to 5
4 Piecewise Function Inputs and Outputs
Exercise 4
Given f(x) = { 2x + 1 if x < 1, x² if x ≥ 1 }, find f(-2), f(1), f(3), and the domain of f(x).
Definition:

Piecewise function: A function defined by different expressions over different intervals

Conditional evaluation: Use appropriate piece based on input value

Piecewise Function
f(x) = {2x + 1 if x < 1, x² if x ≥ 1}
Evaluations
f(-2) = -3, f(1) = 1, f(3) = 9
Step 1: Understand the function definition

f(x) = { 2x + 1 if x < 1

{ x² if x ≥ 1

This means use 2x + 1 when x is less than 1, use x² when x is greater than or equal to 1

Step 2: Find f(-2)

Since -2 < 1, use the first piece: f(x) = 2x + 1

f(-2) = 2(-2) + 1 = -4 + 1 = -3

Step 3: Find f(1)

Since 1 ≥ 1, use the second piece: f(x) = x²

f(1) = 1² = 1

Step 4: Find f(3)

Since 3 ≥ 1, use the second piece: f(x) = x²

f(3) = 3² = 9

Step 5: Determine the domain

All real numbers are covered by the conditions x < 1 and x ≥ 1

There are no restrictions, so the domain is all real numbers

Domain: (-∞, ∞)

Step 6: Verify evaluations

f(-2) = -3: Since -2 < 1, use 2(-2) + 1 = -3 ✓

f(1) = 1: Since 1 ≥ 1, use 1² = 1 ✓

f(3) = 9: Since 3 ≥ 1, use 3² = 9 ✓

f(-2) = -3
f(1) = 1
f(3) = 9
Domain: All real numbers
Final answer:

f(-2) = -3

f(1) = 1

f(3) = 9

Domain: All real numbers (-∞, ∞)

Applied rules:

• Piecewise evaluation: Determine which condition the input satisfies

• Condition checking: Carefully compare input to condition boundaries

• Domain coverage: Ensure all possible inputs are addressed by conditions

5 Function Composition and Input-Output Chains
Exercise 5
Given f(x) = 2x - 1 and g(x) = x + 3, find (f ∘ g)(4) and (g ∘ f)(4). Explain the input-output chain.
Definition:

Function composition: (f ∘ g)(x) = f(g(x)), applying g first, then f

Input-output chain: Tracking how inputs flow through multiple functions

Given
f(x) = 2x - 1, g(x) = x + 3
Composition 1
(f ∘ g)(4) = 9
Composition 2
(g ∘ f)(4) = 9
Step 1: Find (f ∘ g)(4)

(f ∘ g)(4) = f(g(4))

First, find g(4): g(4) = 4 + 3 = 7

Then, find f(7): f(7) = 2(7) - 1 = 14 - 1 = 13

Wait, let me recalculate: f(7) = 2(7) - 1 = 14 - 1 = 13

Step 2: Find (g ∘ f)(4)

(g ∘ f)(4) = g(f(4))

First, find f(4): f(4) = 2(4) - 1 = 8 - 1 = 7

Then, find g(7): g(7) = 7 + 3 = 10

Wait, let me recalculate: g(7) = 7 + 3 = 10

Step 3: Correct calculation for (f ∘ g)(4)

(f ∘ g)(4) = f(g(4))

g(4) = 4 + 3 = 7

f(7) = 2(7) - 1 = 14 - 1 = 13

Step 4: Correct calculation for (g ∘ f)(4)

(g ∘ f)(4) = g(f(4))

f(4) = 2(4) - 1 = 8 - 1 = 7

g(7) = 7 + 3 = 10

Step 5: Trace the input-output chains

For (f ∘ g)(4): Input 4 → g(4) = 7 → f(7) = 13

For (g ∘ f)(4): Input 4 → f(4) = 7 → g(7) = 10

Input-output chain: 4 → 7 → 13 for (f ∘ g)(4)

Input-output chain: 4 → 7 → 10 for (g ∘ f)(4)

Step 6: Verify the process

Composition order matters: (f ∘ g)(x) ≠ (g ∘ f)(x) in general

Here: (f ∘ g)(4) = 13 and (g ∘ f)(4) = 10

Indeed, 13 ≠ 10, confirming order matters in composition

(f ∘ g)(4) = 13
(g ∘ f)(4) = 10
Final answer:

(f ∘ g)(4) = 13

(g ∘ f)(4) = 10

Input-output chains:

(f ∘ g)(4): 4 → 7 → 13

(g ∘ f)(4): 4 → 7 → 10

Applied rules:

• Composition definition: (f ∘ g)(x) = f(g(x))

• Order matters: Function composition is generally not commutative

• Chain evaluation: Work from inside out in compositions

Comprehensive Guide: Understanding Function Relationships
y = f(x)
Input-Output Relationship
Key definitions:

Function: A rule that assigns exactly one output to each input

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

Complete input-output analysis approach:
  1. Identify the function: Recognize the function rule and type
  2. Determine domain: Find restrictions based on function type
  3. Evaluate for inputs: Apply function rule to given inputs
  4. Solve reverse problems: Find inputs when outputs are specified
  5. Analyze behavior: Understand how function transforms inputs
  6. Determine range: Find possible output values
Tip 1: Always verify domain restrictions before evaluating functions.
Tip 2: For radical functions, the expression under the root must be non-negative.
Tip 3: For rational functions, denominators cannot equal zero.
Tip 4: Graph functions to visualize input-output relationships and confirm your analysis.
Common misconceptions: Confusing domain and range, forgetting domain restrictions, assuming all functions are one-to-one.
Memory aids: Domain = inputs (x-values), Range = outputs (y-values), "DR" = "Domain, Range".
Essential formulas and relationships:

• Domain restrictions: √(f(x)) → f(x) ≥ 0, 1/f(x) → f(x) ≠ 0, log(f(x)) → f(x) > 0

• Range finding: Analyze function behavior, critical points, and limits

• Function composition: (f ∘ g)(x) = f(g(x))

• Inverse relationships: If f(a) = b, then a is the input that produces output b

• Function types: Linear (all reals), Quadratic (depends on vertex), Radical (restricted)

Visualization: Input-Output Relationships
Exercise 6: Function Comparison
Compare the input-output relationships of:
f(x) = 2x + 1
g(x) = x²
h(x) = √x

Analysis: Different functions transform inputs differently, affecting their domain and range.

  • Linear functions: Constant rate of change
  • Quadratic functions: Parabolic relationship
  • Radical functions: Restricted domain

Questions & Answers

Question: What's the difference between domain and range? How do I identify them?

Answer: Think of a function as a machine that takes inputs and produces outputs:

Domain (inputs):

  • The set of all possible x-values you can put into the function
  • These are the "allowed" input values
  • Also called the independent variable

Range (outputs):

  • The set of all possible y-values that come out of the function
  • These are the actual output values produced
  • Also called the dependent variable

Memory aid: D(omain) comes before R(ange) alphabetically, just like Inputs come before Outputs!

Example: For f(x) = x², Domain = all real numbers, Range = y ≥ 0 (since x² is always non-negative)

Question: How do I find x when I know f(x)? Like if f(x) = 2x + 3 and f(x) = 11, how do I find x?

Answer: This is called solving for the input when the output is known. Set the function equal to the given output and solve for x:

Given: f(x) = 2x + 3 and f(x) = 11

Set them equal: 2x + 3 = 11

Subtract 3 from both sides: 2x = 8

Divide by 2: x = 4

Verify: f(4) = 2(4) + 3 = 8 + 3 = 11 ✓

The key is to treat this as a regular equation where you solve for the variable that was originally the input.

Always check your answer by substituting back into the original function!

Question: How do I know when a function has domain restrictions? What are the common ones?

Answer: Domain restrictions occur when certain input values would make the function undefined. Here are the most common types:

Square roots: √(expression) → expression ≥ 0

  • For f(x) = √(x - 2), need x - 2 ≥ 0, so x ≥ 2

Fractions: 1/(expression) → expression ≠ 0

  • For f(x) = 1/(x - 3), need x - 3 ≠ 0, so x ≠ 3

Logarithms: log(expression) → expression > 0

  • For f(x) = log(x + 1), need x + 1 > 0, so x > -1

For polynomial functions like f(x) = x² + 3x + 1, there are no domain restrictions - the domain is all real numbers.

Always scan the function for these special operations!

Question: What happens when I compose functions? How do I track the inputs and outputs?

Answer: Function composition creates a chain of transformations. For (f ∘ g)(x) = f(g(x)):

1. Start with input x

2. Apply function g to x, getting g(x)

3. Take the output g(x) and use it as input to function f

4. Get the final output f(g(x))

Think of it as a pipeline: x → g → g(x) → f → f(g(x))

Example: If f(x) = 2x and g(x) = x + 1, then (f ∘ g)(3) = f(g(3)) = f(4) = 8

The output of g(3) = 4 becomes the input to f, giving f(4) = 8.

Order matters: (f ∘ g)(x) is usually different from (g ∘ f)(x).

Question: How do I handle piecewise functions? What if my input falls on a boundary?

Answer: For piecewise functions, you must identify which condition your input satisfies:

Example: f(x) = { x² if x < 2, 3x - 1 if x ≥ 2 }

To find f(1): Since 1 < 2, use f(x) = x², so f(1) = 1² = 1

To find f(2): Since 2 ≥ 2 is true, use f(x) = 3x - 1, so f(2) = 3(2) - 1 = 5

To find f(3): Since 3 ≥ 2, use f(x) = 3x - 1, so f(3) = 3(3) - 1 = 8

For boundary values, look at the inequality signs. If it's ≤ or ≥, the boundary value belongs to that piece.

Always double-check which piece applies to your specific input value!