Problem 44 - Intermediate Algebra

Question

Find each function value if \(f(x)=3 x^{2}-2 .\) See Section 3.2 $$ f(b) $$

Step-by-Step Solution

Answer

\( f(b) = 3b^2 - 2 \)

Step 1: Understand the Function

The given function is defined as \( f(x) = 3x^2 - 2 \). This is a quadratic function, where \( x \) is the variable.

Step 2: Substitute the Variable

To find \( f(b) \), substitute \( b \) into the function in place of \( x \). This turns the function into \( f(b) = 3b^2 - 2 \).

Step 3: Simplify the Expression

Simplify the expression \( 3b^2 - 2 \) to write it in its simplest form. Since there are no like terms to combine or additional simplifications available, \( f(b) = 3b^2 - 2 \) is the final expression.

Key Concepts

Function Evaluation

Understanding function evaluation is like discovering the actual output or value of a function for specific input values. In our exercise, we begin with the function given by the expression \(f(x) = 3x^2 - 2\). When you hear about evaluating a function, think of it as plugging in specific numbers or symbols to replace the variable in the expression. This process transforms the abstract function into a concrete number.

For example, evaluating the function for some value \(b\) means substituting \(b\) in place of \(x\) in the formula. This means we are trying to find \(f(b)\). After substitution, the function may look different but essentially gives us the value solely in terms of \(b\). This is an important concept because it allows us to see how a specific input affects the entire function output.

Algebraic Expressions

Algebraic expressions form the backbone of our function \( f(x) = 3x^2 - 2 \). These expressions are combinations of numbers, variables, and arithmetic operators that represent a particular value. In our scenario, the quadratic term \( 3x^2 \) and the constant \(-2\) make up the expression. Consider the elements:

• Coefficient: The number 3 in front of \(x^2\) is the coefficient. It tells us how many times to multiply \(x^2\).
• Variable: This is the \(x\) in the expression, which may be substituted with any number or symbol.
• Constant: The number -2, which adds or subtracts a fixed amount.

Comprehending these parts is crucial because it allows us to manipulate and evaluate expressions through various operations such as substitution and simplification. Understanding algebraic expressions equips learners with the tools needed to approach and solve complex mathematical problems by breaking them down into digestible parts.

Substitution Method

The substitution method is a key technique for evaluating functions and simplifying algebraic expressions. It's a way to replace variables with specific values, turning an equation into something we can easily compute. In our exercise, we replaced the variable \(x\) with \(b\) in the quadratic function.

• This transforms \(f(x) = 3x^2 - 2\) into \(f(b) = 3b^2 - 2\).
• By substitution, we see the specific impact that \(b\) has on the expression.

Using this method helps us in analyzing and understanding how changes to input variables affect overall outcomes. It also demonstrates that the form of the equation can change, but the fundamental relationships remain the same. Essentially, the substitution method is powerful because it translates abstract mathematical ideas into more concrete computations, enhancing our problem-solving skills and comprehension.