Problem 77 - Intermediate Algebra

Question

Let \(f(x)=-x^{3}+2 x-2\) and \(g(x)=\frac{2-x}{9+x}\) and find each value. $$ f(-1) $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The function given is \( f(x) = -x^3 + 2x - 2 \). This is a cubic function, and we need to evaluate it at a specific point \( x = -1 \).

Step 2: Substitute the Point into the Function

Substitute \( x = -1 \) into the function \( f(x) \):\[f(-1) = -(-1)^3 + 2(-1) - 2\]

Step 3: Calculate the Cube of Negative One

Calculate \((-1)^3\), which is equal to \(-1\):\[(-1)^3 = -1\]

Step 4: Apply Negation to the Cubic Term

Apply the negation to the cubic term:\[-(-1) = 1\]

Step 5: Calculate the Expression

Substitute the results back into the function evaluation:\[f(-1) = 1 + 2(-1) - 2\]This simplifies to:\[f(-1) = 1 - 2 - 2\]

Step 6: Final Calculation

Combine the constants to find the final result:\[f(-1) = 1 - 2 - 2 = -3\]

Key Concepts

Function Evaluation

Evaluating functions means finding the output value of the function for a specific input value. This process is essential in understanding how functions behave at different points. In our exercise, we are tasked with calculating the value of the function \( f(x) = -x^3 + 2x - 2 \) when \( x \) is set to -1.

The process involves several steps:

• First, substitute the given input value into every occurrence of \( x \) in the function expression.
• Perform all arithmetic operations following the order of operations: parentheses, exponents, multiplication/division (from left to right), and addition/subtraction (from left to right).
• The result after all calculations is the output of the function for that specific input value.

In summary, function evaluation is essentially "plugging in" the input and doing the math to find the corresponding output.

Polynomials

Polynomials are mathematical expressions made up of variables, constants, and non-negative integer exponents. These form the backbone of many algebraic operations. In the exercise, the function \( f(x) = -x^3 + 2x - 2 \) is a cubic polynomial because the highest power of \( x \) is 3.

Polynomials have several important characteristics:

• Each term in a polynomial is a product of a constant (or coefficient) and a variable raised to an exponent.
• The degree of a polynomial is determined by the highest exponent present.
• Polynomials are smooth, continuous functions, which means they don't have any abrupt changes, holes, or breaks.
• Cubic polynomials like ours have the capacity to change their direction up to two times, forming an 'S' shape or reversed 'S' shape on a graph.

Understanding these properties helps in analyzing the behavior of polynomial functions across different values of \( x \).

Substitution Method

The substitution method is a fundamental technique often used in algebra to determine the values of a function. It involves replacing a variable with a given value to simplify the equation and solve for another part of it. In this exercise, we used substitution to evaluate the cubic function \( f(x) = -x^3 + 2x - 2 \) by setting \( x = -1 \).

Here's how substitution works in practice:

• Identify the variable to substitute and replace it in the function with the provided numerical value.
• Adapt the mathematical expression by following the correct order of operations to compute the result.
• If necessary, break down complex calculations into simpler steps to ensure accuracy, especially with negative numbers and powers.

Using substitution is like solving a puzzle where you have the pieces and need to place them correctly to see the whole picture.