Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

If \(f(x)=-x^{3}-x^{2}+3\), find \(f(-3), f(0)\), and \(f(2)\).

Step-by-Step Solution

Answer

\(f(-3) = 21\), \(f(0) = 3\), and \(f(2) = -9\).

Step 1: Substitute -3 into the function

To find \(f(-3)\), substitute \(-3\) into the function \(f(x) = -x^3 - x^2 + 3\). Calculate each term: 1. \((-3)^3 = -27\), so \(-(-27) = 27\).2. \((-3)^2 = 9\), so \(-9 = -9\).Thus, \(f(-3) = 27 - 9 + 3 = 21\).

Step 2: Substitute 0 into the function

To find \(f(0)\), substitute \(0\) into the function \(f(x) = -x^3 - x^2 + 3\). Calculate each term:1. \((0)^3 = 0\).2. \((0)^2 = 0\).Thus, \(f(0) = 0 - 0 + 3 = 3\).

Step 3: Substitute 2 into the function

To find \(f(2)\), substitute \(2\) into the function \(f(x) = -x^3 - x^2 + 3\). Calculate each term:1. \((2)^3 = 8\), so \(-8 = -8\).2. \((2)^2 = 4\), so \(-4 = -4\).Thus, \(f(2) = -8 - 4 + 3 = -9\).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output value of a function for a given input value. In this exercise, we are working with a polynomial function, where we need to evaluate it at different input values, namely -3, 0, and 2. To evaluate a function, follow these simple steps:

• Identify the input value you need to substitute into the function.
• Replace every occurrence of the variable (usually represented by x in our example) in the function with the given input value.
• Simplify the resulting expression step by step to find the output.

This technique is helpful in various mathematical and real-world problems, allowing us to understand how changes in input values affect the output of a function. Being thorough with calculations is crucial to avoid any possible errors in function evaluation.

Cubic Functions

Cubic functions are a specific type of polynomial functions where the highest power of the variable is three. The general form of a cubic function is: \[ f(x) = ax^3 + bx^2 + cx + d \]Here, a, b, c, and d are coefficients, and a is not equal to zero. In our exercise, the cubic function is \( f(x) = -x^3 - x^2 + 3 \). Cubic functions are characterized by their unique shapes known as "S-shaped" curves, which often have one inflection point and can have up to three real roots. They are also important in various fields like physics and engineering to model phenomena involving growth or changes. Understanding how to manipulate and evaluate these functions is key to solving a wide range of problems.

Algebraic Expressions

An algebraic expression consists of variables, coefficients, and arithmetic operations like addition, subtraction, multiplication, and division. In the context of evaluating functions, algebraic expressions need to be simplified following the order of operations: parentheses, exponents, multiplication and division (from left to right), and finally, addition and subtraction (also from left to right).In our example expression, \(-x^3 - x^2 + 3\), we first handle the exponent terms after substituting the variable. For instance, to find \( f(-3) \), replace \( x \) with \(-3\) and perform the calculations:

• Calculate \((-3)^3\) to get \(-27\), apply the negative sign for -(-27) = 27.
• Next, \((-3)^2\) is 9, the negative makes it -9.
• Combine all terms to simplify: \(27 - 9 + 3\).

Algebraic expressions allow you to systematically solve problems involving unknown quantities by breaking down complex operations into manageable steps.