Problem 15 - Elementary Algebra

Question

State the restrictions and simplify. Given \(f(x)=x-3 x 2+9,\) find \(f(-3), f(0),\) and \(f(3)\).

Step-by-Step Solution

Answer

f(-3) = -21, f(0) = 9, f(3) = -15.

Step 1: Identify the Function

The given function is \(f(x) = x - 3x^2 + 9\). We need to evaluate this function at different values of \(x\) to find \(f(-3), f(0),\) and \(f(3)\).

Step 2: Evaluate f(-3)

Substitute \(-3\) into the function: \[ f(-3) = (-3) - 3(-3)^2 + 9 \]Calculate step-by-step: 1. Calculate \((-3)^2\) to get \(9\).2. Multiply \(9\) by \(3\) to get \(27\).3. Substitute these values: \[ f(-3) = -3 - 27 + 9 \]4. Compute to get: \[ f(-3) = -21 \]

Step 3: Evaluate f(0)

Substitute \(0\) into the function: \[ f(0) = 0 - 3(0)^2 + 9 \]The calculation simplifies immediately because all terms apart from the constant disappear: \[ f(0) = 9 \]

Step 4: Evaluate f(3)

Substitute \(3\) into the function: \[ f(3) = 3 - 3(3)^2 + 9 \]Calculate step-by-step: 1. Calculate \((3)^2\) to get \(9\).2. Multiply \(9\) by \(3\) to get \(27\).3. Substitute these values: \[ f(3) = 3 - 27 + 9 \]4. Compute to get: \[ f(3) = -15 \]

Key Concepts

Function Evaluation

Evaluating a polynomial function means finding the value of the function at a certain point. In other words, we want to know what the function equals when we substitute a specific number for the variable. In this exercise, we are given a function, \(f(x) = x - 3x^2 + 9\), and we want to find the value of this function for \(f(-3)\), \(f(0)\), and \(f(3)\). Function evaluation is essential for understanding how a function behaves at specific points. This process can reveal critical information about the function, such as its tendency to increase or decrease, and identifying any values where it might have specific characteristics, like achieving a maximum or minimum. By evaluating the function at the given points, we build a clearer picture of how it operates. Always ensure you understand the form of the function before evaluation, noting each term and its role, like constants, coefficients, and variable terms.

Substitution

Substitution is the technique of replacing the variable in the function with the number you want to evaluate. In the function \(f(x) = x - 3x^2 + 9\), to find \(f(-3)\), \(f(0)\), and \(f(3)\), we'll replace \(x\) with \(-3\), \(0\), and \(3\), respectively.Substitution follows a systematic approach:

• Identify the variable in the function.
• Replace this variable everywhere in the expression with the given number.
• Maintain the order of operations: parentheses, exponents, multiplication/division, and addition/subtraction (PEMDAS).

By substituting \(-3\) into the original equation, we replace every \(x\) with \(-3\). The new equation becomes \((-3) - 3(-3)^2 + 9\). Performing similar steps for \(x = 0\) and \(x = 3\) gives us the equations for each respective evaluation.

Simplification

Simplification is the method of reducing expressions to their simplest form. This involves combining like terms, performing arithmetic operations, and following algebraic rules. For instance, after substituting values into \(f(x) = x - 3x^2 + 9\), one must simplify to find \(f(x)\) at each evaluated point.After substituting \(-3\), in \((-3) - 3(-3)^2 + 9\), operations include:

• Calculating exponents (\((-3)^2 = 9\)).
• Multiplying results (\(3 \times 9 = 27\)).
• Finally, add and subtract the terms: \(-3 - 27 + 9\) simplifies to \(-21\).

In practice, simplification ensures that the answer is neat, concise, and easy to interpret. It also verifies any patterns or trends in the values obtained, leading us to a deeper understanding of the function's properties.

Restrictions

In the context of polynomial functions, restrictions refer to constraints on the variable that can impact the function’s domain or further solutions. Though basic polynomial functions generally have fewer restrictions, understanding the concept is valuable for complex expressions.In the given polynomial \(f(x) = x - 3x^2 + 9\), all real numbers are permissible since no division by zero or square roots of negative numbers occur. However, restrictions are still a vital consideration in mathematical analysis and function evaluation.For more complex situations:

• Watch for denominators (no division by zero).
• Be cautious with square roots (only take square roots of non-negative numbers in real number context).
• Consider restrictions set by practical scenarios (like context-based constraints in applied contexts).

By understanding potential restrictions, one can ensure accurate evaluations and interpretations throughout mathematical problem-solving.