Problem 33 - Algebra and Trigonometry

Question

Use the function to evaluate the indicated expressions and simplify. $$ f(x)=x+4 ; \quad f\left(x^{2}\right),(f(x))^{2} $$

Step-by-Step Solution

Answer

\(f(x^2) = x^2 + 4; \quad (f(x))^2 = x^2 + 8x + 16\).

Step 1: Evaluate \(f(x^2)\)

To evaluate \(f(x^2)\), substitute \(x^2\) into the expression for \(f(x)\). Since \(f(x) = x + 4\), replace \(x\) with \(x^2\) to get \(f(x^2) = x^2 + 4\).

Step 2: Evaluate \((f(x))^2\)

To find \((f(x))^2\), first determine \(f(x)\), which is \(x + 4\). Square the result: \( (x + 4)^2 \). Use the formula \((a + b)^2 = a^2 + 2ab + b^2\) to expand it: \( (x + 4)^2 = x^2 + 2 \cdot x \cdot 4 + 4^2 = x^2 + 8x + 16 \).

Key Concepts

Understanding Polynomial Functions

Polynomial functions are a key concept in algebra that involve expressions consisting of variables and coefficients. These variables are usually raised to whole number powers. For example, a simple polynomial function can look like this: \(f(x) = x + 4\). Here, \(x\) is the variable, very much like an undefined number that can be varied, and \(4\) is a constant coefficient added to \(x\).
When dealing with polynomial functions, you often work with different expressions to either evaluate or simplify them. A polynomial function's structure is flexible and can contain terms like \(x^2, x^3\), and so on. The task often involves substituting values and simplifying using algebraic rules, which leads to the next sections that explain this process in more detail.

Breaking Down Function Expressions

A function expression, like \(f(x) = x + 4\), essentially gives you a rule for what to do with any input value \(x\). When you want to evaluate this for a new expression of \(x\), such as \(x^2\), the process involves simply putting this value in place of every \(x\) in the function.

This is a two-pronged approach in our problem:

• First, evaluate \(f(x^2)\) by replacing \(x\) with \(x^2\) in \(f(x)\). You get \(f(x^2) = x^2 + 4\).
• Then, evaluate \((f(x))^2\), where you find \(f(x)\) first, which is \(x + 4\), and then square it. The expression becomes \((x + 4)^2\).

These steps illustrate how function expressions allow for dynamic evaluations based on the input taken by the variable \(x\). This flexibility is crucial for learning how to manipulate and understand diverse algebraic expressions.

Mastering Algebraic Simplification

Algebraic simplification is the art of reducing expressions to their simplest form for ease of use and interpretation. In our task, after evaluating the function expressions, we must simplify them.
Let’s focus on \((f(x))^2\), which results into \((x+4)^2\). This can be expanded using the formula \((a + b)^2 = a^2 + 2ab + b^2\). Here:

• \(a = x\) and \(b = 4\), leading us to compute \(x^2 + 2 \cdot x \cdot 4 + 4^2\).
• This simplifies to \(x^2 + 8x + 16\).

Each expanded term combines through basic arithmetic into a more straightforward polynomial form. Simplifying algebraic expressions is a critical skill that helps in solving equations, understanding relationships, and verifying identities in both pure and applied mathematics.