Function Notation | Solved Exercises

Solutions: Exercises 1 to 10

1 Basic function evaluation

Exercise 1

If f(x) = 2x + 3, find f(5).

Difficulty: Beginner Time: ~2 minutes Skills: Function Evaluation

Definition:

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

Note: f(x) is read as "f of x" and represents the y-value or output corresponding to input x.

Step-by-step evaluation method:

Identify the function rule (the expression after f(x) = )
Replace every instance of x with the given input value
Simplify the expression using order of operations
Record the result as the function output

Function Rule

f(x) = 2x + 3

Input Value

x = 5

Result

f(5) = 13

Step 1: Identify the function

f(x) = 2x + 3

Step 2: Substitute x = 5

f(5) = 2(5) + 3

Step 3: Calculate

f(5) = 10 + 3 = 13

f(5) = 13

Final answer:

f(5) = 13

Applied rules:

• Substitution: Replace x with the given input value

• Order of operations: Perform operations in correct sequence

• Simplification: Calculate the final result accurately

• Practice Tip: Always perform operations in the correct order: parentheses, exponents, multiplication/division, addition/subtraction

Related Examples:

If g(x) = x + 7, then g(3) = 3 + 7 = 10
If h(x) = 4x - 1, then h(2) = 4(2) - 1 = 7
If p(x) = -x + 5, then p(-2) = -(-2) + 5 = 7

Quick Tips:

Think of f(x) as a machine that takes input x and produces output f(x)
Always substitute the input value in place of every occurrence of x
Be careful with negative inputs and negative coefficients

Frequently Asked Questions:

Q: What does f(x) mean?
A: f(x) means "the value of function f when x is the input" - it's another way of writing y.

Q: Can I have multiple functions like f(x), g(x), h(x)?
A: Yes, different letters represent different functions: f(x), g(x), h(x), etc.

2 Function notation substitution

Exercise 2

If g(x) = x² - 4x + 1, find g(-2).

Difficulty: Beginner Time: ~3 minutes Skills: Function Evaluation with Powers

Definition:

Function evaluation: Replacing the variable in a function with a specific value

Note: When substituting negative values, pay special attention to the signs of each term.

Step-by-step substitution method:

Identify the function rule: g(x) = x² - 4x + 1
Replace every x with the given input value (-2)
Calculate each term separately, being careful with signs
Combine the terms to get the final result

Function Rule

g(x) = x² - 4x + 1

Input Value

x = -2

Result

g(-2) = 13

Step 1: Identify the function

g(x) = x² - 4x + 1

Step 2: Substitute x = -2

g(-2) = (-2)² - 4(-2) + 1

Step 3: Calculate each term

(-2)² = 4, -4(-2) = 8, +1 = 1

Step 4: Combine terms

g(-2) = 4 + 8 + 1 = 13

g(-2) = 13

Final answer:

g(-2) = 13

Applied rules:

• Sign handling: Be careful with negative inputs, especially when raising to powers

• Order of operations: Calculate exponents first, then multiplication, then addition/subtraction

• Systematic calculation: Work through each term separately

• Practice Tip: Remember (-a)² = a², but (-a)³ = -a³

Related Examples:

If h(x) = x² + 3x - 2, then h(-1) = 1 - 3 - 2 = -4
If p(x) = -x² + 2x + 5, then p(3) = -9 + 6 + 5 = 2
If q(x) = x² - 6x + 9, then q(3) = 9 - 18 + 9 = 0

Quick Tips:

Pay special attention to signs when substituting negative values
Calculate each term separately to avoid sign errors
Remember that squaring a negative number gives a positive result

Frequently Asked Questions:

Q: What if the function has multiple terms with x?
A: Substitute the input value for every occurrence of x, then calculate each term separately.

Q: How do I handle negative inputs with exponents?
A: (-a)² = a² (positive), but (-a)³ = -a³ (negative). The result depends on whether the exponent is even or odd.

3 Evaluating linear functions

Exercise 3

If f(x) = 3x - 7, find f(0), f(2), and f(-1).

Difficulty: Intermediate Time: ~4 minutes Skills: Multiple Function Evaluations

Definition:

Linear function: A function of the form f(x) = mx + b, where m and b are constants

Note: Linear functions have a constant rate of change and produce straight lines when graphed.

Step-by-step multiple evaluation method:

Identify the function rule: f(x) = 3x - 7
For each input value, substitute into the function rule
Follow order of operations: multiplication, then addition/subtraction
Calculate and record each output value

Function Rule

f(x) = 3x - 7

f(0)

3(0) - 7 = -7

f(2)

3(2) - 7 = -1

f(-1)

3(-1) - 7 = -10

Step 1: Evaluate f(0)

f(0) = 3(0) - 7 = 0 - 7 = -7

Step 2: Evaluate f(2)

f(2) = 3(2) - 7 = 6 - 7 = -1

Step 3: Evaluate f(-1)

f(-1) = 3(-1) - 7 = -3 - 7 = -10

f(0) = -7, f(2) = -1, f(-1) = -10

Final answer:

f(0) = -7, f(2) = -1, f(-1) = -10

Applied rules:

• Substitution: Replace x with the given input value

• Order of operations: Perform operations in correct sequence

• Sign handling: Be careful with negative inputs

• Practice Tip: Work systematically through each value to avoid errors

Related Examples:

If g(x) = 2x + 5, then g(0) = 5, g(1) = 7, g(-1) = 3
If h(x) = -x + 4, then h(0) = 4, h(2) = 2, h(-3) = 7
If p(x) = 4x - 3, then p(0) = -3, p(1) = 1, p(-2) = -11

Quick Tips:

When x = 0, the result is just the constant term (b in f(x) = mx + b)
Work through each evaluation systematically to avoid confusion
Check your arithmetic carefully, especially with negative values

Frequently Asked Questions:

Q: What happens when x = 0 in a linear function?
A: The result is just the constant term. For f(x) = mx + b, f(0) = b.

Q: How do I check my answers?
A: Substitute the input back into the original function and verify the calculation.

Solutions: Exercises 4 to 5

4 Quadratic function evaluation

Exercise 4

If h(x) = x² + 2x - 3, find h(3) and h(-3).

Definition:

Quadratic function: A function of the form f(x) = ax² + bx + c, where a ≠ 0

Note: Quadratic functions have degree 2 and produce parabolic graphs when plotted.

Step-by-step quadratic evaluation method:

Identify the function rule: h(x) = x² + 2x - 3
For each input value, substitute into the function rule
Calculate each term separately: x², 2x, and -3
Combine the terms to get the final result

Function Rule

h(x) = x² + 2x - 3

h(3)

9 + 6 - 3 = 12

h(-3)

9 - 6 - 3 = 0

Step 1: Evaluate h(3)

h(3) = (3)² + 2(3) - 3 = 9 + 6 - 3 = 12

Step 2: Evaluate h(-3)

h(-3) = (-3)² + 2(-3) - 3 = 9 - 6 - 3 = 0

Step 3: Verify calculations

h(3) = 9 + 6 - 3 = 12 ✓

h(-3) = 9 - 6 - 3 = 0 ✓

h(3) = 12, h(-3) = 0

Final answer:

h(3) = 12 and h(-3) = 0

Applied rules:

• Quadratic evaluation: Calculate each term separately (x², bx, c)

• Sign handling: Pay attention to signs when substituting negative values

• Order of operations: Exponents first, then multiplication, then addition/subtraction

• Practice Tip: Remember that (-a)² = a² regardless of the sign of a

Related Examples:

If f(x) = x² - 4, then f(2) = 0 and f(-2) = 0
If g(x) = x² + x + 1, then g(0) = 1 and g(-1) = 1
If p(x) = 2x² - 3x + 1, then p(1) = 0 and p(-1) = 6

Quick Tips:

Quadratic functions often have symmetry around their vertex
Squaring negative numbers always gives positive results
Pay attention to the signs of each term when substituting

Frequently Asked Questions:

Q: What if the quadratic has a coefficient for x²?
A: Multiply the coefficient by the squared term. For f(x) = 2x² + x - 1, f(3) = 2(9) + 3 - 1 = 20.

Q: Can quadratic functions have the same output for different inputs?
A: Yes, due to their symmetric nature, quadratics can have f(a) = f(b) for a ≠ b.

5 Function operations

Exercise 5

If f(x) = 2x + 1 and g(x) = x - 3, find f(4) + g(4).

Definition:

Function operations: Performing arithmetic operations on function values

Note: We can add, subtract, multiply, or divide function values just like regular numbers.

Step-by-step operation method:

Evaluate each function separately at the given input
Perform the arithmetic operation on the results
Combine to get the final answer

Function Rules

f(x) = 2x + 1, g(x) = x - 3

f(4)

2(4) + 1 = 9

g(4)

4 - 3 = 1

f(4) + g(4)

9 + 1 = 10

Step 1: Evaluate f(4)

f(4) = 2(4) + 1 = 8 + 1 = 9

Step 2: Evaluate g(4)

g(4) = 4 - 3 = 1

Step 3: Add the results

f(4) + g(4) = 9 + 1 = 10

f(4) + g(4) = 10

Final answer:

f(4) + g(4) = 10

Applied rules:

• Separate evaluation: Calculate each function value independently

• Arithmetic operations: Perform operations on the results

• Order of operations: Follow proper sequence when evaluating

• Practice Tip: Evaluate each function completely before performing operations

Related Examples:

If f(x) = x + 2 and g(x) = x - 1, then f(3) - g(3) = 5 - 2 = 3
If p(x) = 2x and q(x) = x + 1, then p(2) × q(2) = 4 × 3 = 12
If r(x) = x² and s(x) = x, then r(3) ÷ s(3) = 9 ÷ 3 = 3

Quick Tips:

Evaluate each function separately before performing operations
Keep track of which function you're evaluating at each step
Double-check your arithmetic after evaluating each function

Frequently Asked Questions:

Q: Can I combine functions symbolically first?
A: Yes, you could define (f+g)(x) = f(x) + g(x) = (2x+1) + (x-3) = 3x-2, then evaluate (f+g)(4) = 10.

Q: What if I need to multiply function values?
A: Same approach: evaluate each function separately, then multiply the results.

Questions & Answers

FunctionLearner

Middle School Level

Question: I'm confused about f(x) notation. Is it multiplication? What does it mean?

ProfessorMath

Mathematics Expert - PhD

Answer: Great question! f(x) is NOT multiplication. It's function notation:

f(x) is read as "f of x" and represents the output of function f when x is the input
It's a way to name the output value, similar to how we might write y = 2x + 1
f(x) = 2x + 1 means the same thing as y = 2x + 1, but emphasizes that y depends on x
The letter f is just the name of the function; we could also use g(x), h(x), etc.

For example, if f(x) = 2x + 1, then f(3) = 2(3) + 1 = 7. This means when x = 3, the function outputs 7.

Function notation helps us work with multiple functions simultaneously and clearly express relationships between inputs and outputs.

CompoMaster

Grade 8 Student

Question: How do I evaluate f(g(x))? I don't understand the order.

TutorPro

Master in Mathematical Education

Answer: Function composition f(g(x)) follows an inside-out approach:

Start with the innermost function g(x) and evaluate it first
Take the result from g(x) and use it as the input for function f
Finally, evaluate f with this new input

For example, if f(x) = 2x + 1 and g(x) = x - 3, to find f(g(5)):
1. First evaluate g(5) = 5 - 3 = 2
2. Then use this result as input for f: f(2) = 2(2) + 1 = 5
Therefore, f(g(5)) = 5

Think of it as a chain: input → g → result of g → f → final result.

WordProbSolver

Middle School Student

Question: How do I know what the variables represent in word problems?

MathMentor

Certified Mathematics Teacher

Answer: Identifying variable meanings in word problems is crucial:

Look for explicit statements like "where x is the number of hours" or "C(d) represents cost in dollars"
The input variable (usually x) typically represents the independent quantity that can be controlled or measured
The function output (f(x)) typically represents the dependent quantity that results from the input
Pay attention to units mentioned in the problem (dollars, hours, items, etc.)

For example, in "The cost of renting a car is modeled by C(d) = 40d + 25":
- d represents the number of days rented (input/independent variable)
- C(d) represents the total cost in dollars (output/dependent variable)
- The function shows how cost depends on the number of days

Always state what each variable represents before solving the problem.

Detailed Summary: Function Notation

Definitions and Concepts

Function Notation: The symbolic way to represent functions using f(x), g(x), h(x), etc., where f is the function name and x is the input variable. This notation indicates that the function f takes an input x and produces an output.

Function Evaluation: The process of finding the output value of a function for a specific input value. This involves substituting the input value for x in the function rule and simplifying the expression.

Function Composition: The operation of applying one function to the result of another function, denoted as f(g(x)). This creates a new function by combining two or more functions.

Core Rules and Principles

Substitution Rule: When evaluating f(a), replace every occurrence of x in the function rule with the value a, then simplify using order of operations.

Composition Order: In f(g(x)), evaluate the inner function g(x) first, then use its result as input for the outer function f.

Variable Substitution: When substituting expressions for x, treat the entire expression as a single unit and apply the function rule accordingly.

Step-by-Step Methods

Basic Function Evaluation: 1) Identify the function rule, 2) Substitute the input value for x, 3) Simplify using order of operations, 4) State the result.

Function Composition: 1) Identify both functions, 2) Evaluate the inner function first, 3) Use the result as input for the outer function, 4) Simplify the final expression.

Function Rule Determination: 1) Examine given input-output pairs, 2) Look for patterns in the differences between outputs, 3) Determine the function type (linear, quadratic, etc.), 4) Find the specific rule parameters.

Examples (Simple to Advanced)

Simple: If f(x) = 2x + 3, then f(5) = 2(5) + 3 = 13

Intermediate: If g(x) = x² - 4x + 1, then g(-2) = (-2)² - 4(-2) + 1 = 4 + 8 + 1 = 13

Advanced: If f(x) = 2x + 1 and g(x) = x - 3, then f(g(x)) = f(x - 3) = 2(x - 3) + 1 = 2x - 6 + 1 = 2x - 5

Tips, Tricks, and Common Pitfalls

Tips: Always work from inside out for compositions, pay attention to signs when substituting negative values, and verify your results by checking calculations.

Tricks: When x = 0 in a linear function f(x) = mx + b, the result is always b; remember that (-a)² = a² for any value of a.

Common Pitfalls: Confusing function notation with multiplication, forgetting to substitute entire expressions, incorrect order in compositions, and sign errors when working with negatives.

Key Notes for Memorization

Memory Aids: "f of x" means "f depends on x"; for compositions, think "do the inside function first"; f(x) is just a fancy way of saying "y equals something with x."

Core Concept: Functions are machines that take inputs and produce outputs according to a specific rule. Function notation is a compact way to express this relationship.

Connection: Function notation connects to real-world applications where one quantity depends on another, making it a powerful tool for modeling situations.

Student-Friendly Explanations

Think of a function as a special kind of machine. You put something in (the input x), the machine does something to it according to the function rule, and you get something out (the output f(x)). Function notation is just a way to label this machine and describe what it does.

Function evaluation is like feeding the machine specific inputs to see what comes out. Function composition is like connecting two machines together so the output of the first machine becomes the input of the second machine.

Function Notation Glossary

Function Notation

The symbolic representation f(x) that denotes the output of function f when x is the input. Read as "f of x".

Function Evaluation

The process of finding the output value of a function for a specific input value by substituting the input into the function rule.

Function Composition

The operation of applying one function to the result of another function, denoted as f(g(x)).

Input Variable

The independent variable (usually x) that is substituted into the function rule to obtain an output value.

Output Value

The dependent value (f(x)) that results from substituting an input value into the function rule.

Function Notation in Grade 8 - Mathematics - Exercises with solutions

Key Takeaways

Questions & Answers

Detailed Summary: Function Notation

Definitions and Concepts

Core Rules and Principles

Step-by-Step Methods

Examples (Simple to Advanced)

Tips, Tricks, and Common Pitfalls

Key Notes for Memorization

Student-Friendly Explanations

Function Notation Glossary

Function Notation Educational Team