Problem 27 - 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

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

Step 1: Understanding the function f(x)

The function given is \( f(x) = x + 4 \). This means for any input \( x \), the output will be \( x + 4 \). We'll use this definition to evaluate the expressions.

Step 2: Evaluate f(x²) using f(x)

To find \( f(x^2) \), replace \( x \) in \( f(x) = x + 4 \) with \( x^2 \). This gives \( f(x^2) = x^2 + 4 \).

Step 3: Evaluate (f(x))²

First, find \( f(x) \), which we've established is \( x + 4 \). Now, compute \((f(x))^2\):\[ (f(x))^2 = (x + 4)^2 \]We expand this by using the formula \((a + b)^2 = a^2 + 2ab + b^2\):\[ (x + 4)^2 = x^2 + 2\cdot x \cdot 4 + 4^2 = x^2 + 8x + 16 \]

Step 4: Simplify the expressions

Both expressions are already in their simplest form:- \( f(x^2) = x^2 + 4 \)- \( (f(x))^2 = x^2 + 8x + 16 \)

Key Concepts

Function Notation

Function notation is a way to denote a specific output for a given input in a function. It's like a rule or a machine where you input a value, and the function processes it according to its formula, then outputs a result. In the function notation \( f(x) = x + 4 \), \( f \) is the name of the function and \( x \) is the variable that you replace with an actual number to get a result.
When you see \( f(x) \), read it as "the function \( f \) of \( x \)." It tells you that whatever comes after \( f \) in parentheses is the input value. This is a crucial part of understanding functions, as it keeps track of what variable you're working with and how it's transformed by the function.

Substitution in Functions

Substitution in functions is the method of replacing the variable inside the function with a specific value or expression. For example, if you have the function \( f(x) = x + 4 \), to find \( f(x^2) \), you substitute \( x^2 \) for \( x \) in the function. This changes the entire expression inside \( f \).

• You start with \( f(x) = x + 4 \).
• To find \( f(x^2) \), replace \( x \) with \( x^2 \).

By substituting \( x^2 \), you create a new expression: \( f(x^2) = x^2 + 4 \). Remember, substitution changes only where the variable \( x \) is present in the function formula.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition or multiplication). They form the building blocks of algebra and functions.
In our example, \( f(x) = x + 4 \) is a simple algebraic expression. It is made up of:

• Variable: \( x \)
• Constant: 4
• Operation: Addition

Algebraic expressions can be more complex, as seen in the outcome for the function application. For example, substituting \( x^2 \) into \( f(x) \) turns the expression into \( x^2 + 4 \), maintaining its algebraic form with more variable complexity.

Algebraic Manipulation

Algebraic manipulation involves rewriting expressions in different forms using mathematical operations. It helps simplify or expand expressions for clearer understanding or easier computation.
In our example, when evaluating \( (f(x))^2 \), we employ algebraic manipulation to expand \( (x + 4)^2 \) using the formula \((a + b)^2 = a^2 + 2ab + b^2\).

• Start with \( (x + 4)^2 \).
• Expand it to \( x^2 + 2 \cdot x \cdot 4 + 4^2 \).
• Result: \( x^2 + 8x + 16 \).

This transformation simplifies the calculation and provides a clearer view of the expression's structure. Algebraic manipulation is a powerful tool that allows flexibility in handling mathematical expressions.