Question

If \(f(x)=8 x+2\) find \(f(f(x))\).

Short answer

Answer: The composition of the function \(f(x) = 8x + 2\) with itself, \(f(f(x))\), is equal to \(64x + 18\).

Step-by-step solution

Identify the function f(x)

The function is given: \(f(x) = 8x + 2\)

Write down f(f(x)) in terms of f(x)

To find \(f(f(x))\), replace \(x\) in the initial function with \(f(x)\). \(f(f(x)) = f(8x + 2)\)

Replace f(x) in f(f(x)) with the expression in terms of x

Now, we need to replace the \(f(x)\) in \(f(f(x))\) expression with the initial function expression: \(f(8x + 2) = 8(8x + 2) + 2\)

Simplify the expression

Let's simplify the expression to get the final result: \(8(8x + 2) + 2 = 64x + 16 + 2\) \(f(f(x)) = 64x + 18\) So, the composition of the function \(f(x)\) with itself, \(f(f(x))\), is equal to \(64x + 18\).

Key concepts

Algebraic Functions

Algebraic functions are mathematical expressions made up of constants, variables, and operations such as addition, subtraction, multiplication, and division. An everyday example would be a line function like \(f(x) = 8x + 2\). The term "algebraic functions" signifies any formula that can be expressed using algebraic operations.

• Linear Functions: These are straightforward linear relationships expressed as \(y = mx + b\).
• Quadratic Functions: Commonly represented as \(ax^2 + bx + c\), these involve squaring the variable.
• Polynomial Functions: This broader category includes any finite sum of constant coefficients and variables raised to whole number exponents.

Understanding these functions often involves evaluating them at specific values of \(x\) to determine the outcome or analyzing the behavior as \(x\) tends towards infinity.
Each function type has its own specific properties and graphical representation, making them a crucial part of calculus and advanced mathematics.

Mathematical Notation

Mathematical notation is a language used to symbolize abstract concepts and relationships in mathematics. It simplifies complex ideas into digestible symbols and formulae, allowing easy communication of mathematical thought. In our exercise, \(f(x)\) signifies a function of \(x\) defined by \(8x + 2\).

• Symbols: Letters such as \(x\) or \(f\) serve as placeholders. They represent variables or specific operations.
• Operators: Symbols like "+", "-", "*", and "/" denote operations that are performed on the numbers or variables.
• Function Notation: The format \(f(x)\) denotes 'a function of \(x\)', making it clear that the output is dependent on the variable \(x\).

The abstraction of these notations allows mathematicians to generalize formulating methods for solving equations and express these equations universally, irrespective of the specific values involved.

Function Evaluation

Function evaluation involves determining the output of a function for a specific input. It's a key skill in understanding how functions operate and how they can be applied to real-world problems.
In our exercise, to evaluate \(f(f(x))\), we substitute \(f(x)\) into the function itself:

• Substitution Process: Start with the function \(f(x) = 8x + 2\).
• Evaluate \(f(x)\) within itself: Replace the \(x\) in \(f(x)\) with \(8x + 2\) to get \(f(f(x)) = f(8x + 2)\).
• Perform arithmetic operations: Calculate \(8(8x + 2) + 2\), simplifying to \(64x + 18\).

After each evaluation, simplification is necessary to reach the cleanest form of the result. This ensures that the function composition is neatly expressed and easily understandable. Correct evaluation enables deeper insights into how altering inputs affect outputs, which is essential in modeling scenarios and predicting outcomes.