Problem 78 - Intermediate Algebra

Question

Let \(f(x)=-x^{3}+2 x-2\) and \(g(x)=\frac{2-x}{9+x}\) and find each value. $$ f(10) $$

Step-by-Step Solution

Answer

\(f(10) = -982\)

Step 1: Identify the function for computation

The function given in the problem is \( f(x) = -x^3 + 2x - 2 \). We need to calculate \( f(10) \).

Step 2: Substitute the value of x into the function

Substitute \( x = 10 \) into the function \( f(x) = -x^3 + 2x - 2 \) to find \( f(10) \).

Step 3: Calculate the cube of the input value

Calculate \( 10^3 \), which is \( 1000 \). This is the first term we need for substitution in the function.

Step 4: Compute the first term with the negative sign

Multiply \( 1000 \) by \(-1\) to get \(-1000 \). This is part of the function \( f(x) = -x^3 + 2x - 2 \).

Step 5: Calculate the second component

Compute \( 2 \times 10 \), which equals \( 20 \). This forms the second term of the equation.

Step 6: Combine the terms

Add the calculated terms: \(-1000 + 20 - 2\). This gives us the result of the function.

Step 7: Compute the final result

Calculate \(-980 - 2\), which is \(-982\). This completes the computation for \( f(10) \).

Key Concepts

Polynomial Functions

Polynomial functions are a type of mathematical expression composed of variables, coefficients, and exponents. The general form of a polynomial function can be expressed as: \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \]where:

• \( a_n, a_{n-1}, ..., a_1, a_0 \) are coefficients or constants that can be any real number.
• \( n \) is a non-negative integer, indicating the highest degree of the polynomial.

An important feature of polynomial functions is their continuity and differentiability, which makes them useful in modeling various real-world situations.

For example, in our exercise, we have the polynomial function \( f(x) = -x^3 + 2x - 2 \). Here, the highest degree is 3, making it a cubic function. Understanding the structure of polynomial functions helps us to easily compute their values at specific points through substitution.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They are fundamental in forming equations and analyzing relationships between variables. A basic algebraic expression might look like:\[ 3x + 4y - 7 \]In algebraic expressions, each component serves a purpose:

• Numerical coefficients, like 3 and 4, determine the weight of the variable they are multiplied by.
• Variables, such as \( x \) and \( y \), represent unknown values that can vary.
• Constants, such as -7, are fixed values that do not change.

In our exercise, the expression \( -x^3 + 2x - 2 \) consists of three terms, where each term is either a constant or involves a variable. Understanding these components is essential for evaluating expressions at given values.

Mathematical Substitution

Substitution is a common method used to evaluate functions and expressions. It involves replacing a variable with a given number. This technique is crucial when solving problems that require determination of specific values of a function.

For instance, given the function \( f(x) = -x^3 + 2x - 2 \), we might want to find the output when \( x = 10 \). To do so, we substitute \( 10 \) where \( x \) appears in the equation, performing step-by-step arithmetic operations to simplify it:

• Compute \( 10^3 \), which is 1000.
• Multiply this result by -1 to get -1000.
• Calculate \( 2 \times 10 \) which gives 20.
• Add all computed terms following the order of operations: \(-1000 + 20 - 2\).
• This yields \(-982\), the result of the substitution.

Substitution simplifies the process of evaluating complex functions, turning them into straightforward arithmetic procedures.