Problem 14 - Algebra 2 and Trigonometry

Question

\(\operatorname{In} 9-14, y=f(x) .\) Find \(f(-3)\) for each function. \(\mathrm{f}(x)=|x+2|\)

Step-by-Step Solution

Answer

The value of \(f(-3)\) is 1.

Step 1: Identify the Function Composition

The given function is \(f(x) = |x + 2|\). We need to find \(f(-3)\), which means we need to evaluate the function when \(x = -3\).

Step 2: Substitute \(x = -3\) into the Function

Substitute \(-3\) for \(x\) in the function: \(f(-3) = |-3 + 2|\).

Step 3: Simplify Inside the Absolute Value

Calculate \(-3 + 2 = -1\). So, the expression inside the absolute value is \(-1\).

Step 4: Apply the Absolute Value

Take the absolute value of \(-1\), which is \(|-1| = 1\).

Step 5: Conclude the Evaluation

The final result of evaluating \(f(-3) = |x + 2|\) is \(1\).

Key Concepts

Function Evaluation

Function evaluation is the process of determining the output of a function for a specific input value. When we talk about evaluating a function like \( f(x) = |x + 2| \), we focus on finding out what the function equals when we plug in different numbers for \( x \).

• We start by identifying the exact form of the given function. Here, the function is \( f(x) = |x + 2| \).
• The function tells us to take whatever number you have for \( x \), add 2 to it, and then determine the absolute value of the result.

By understanding this, we can accurately compute the value of the function for different inputs.

Substituting Variables

Substituting variables is all about taking one specific value and putting it in place of a variable in a mathematical expression or function. In this exercise, it means we need to substitute \( x = -3 \) into the function \( f(x) = |x + 2| \).

• To do this, replace every \( x \) in the function with \( -3 \). This gives us \( f(-3) = |-3 + 2| \).
• This step is crucial because it transforms a general problem into something concrete we can solve.

Working through this substitution process helps us see how the input directly affects the outcome.

Simplification of Expressions

Simplification of expressions often involves making complex mathematical parts easier to understand or solve. With absolute value functions, simplification is very important.

• First, perform any arithmetic inside the absolute value: Calculate \(-3 + 2\), which equals \(-1\).
• The absolute value function \(|x|\) turns any negative number inside it into positive, so \(|-1|\) becomes \(1\).

This simplification process involves both arithmetic calculations and understanding that absolute value converts all quantities to their non-negative counterparts. Through simplification, the function \( f(-3) \) evaluates to \(1\) in a clear and straightforward way.