Problem 53 - Elementary and Intermediate Algebra

Question

Find \(g(2)\) and \(g(3) .\) See Example 4. $$ g(x)=-x^{3}+x $$

Step-by-Step Solution

Answer

\(g(2) = -6\), \(g(3) = -24\)

Step 1: Understanding the Function

The function given is \(g(x) = -x^3 + x\). This means for any input \(x\), we first cube it, then negate the result, and finally add \(x\) itself.

Step 1: Calculate \(g(2)\)

To find \(g(2)\), substitute \(x = 2\) into the function. So, we have:\[g(2) = -(2)^3 + 2\]

Step 2: Simplify \(g(2)\)

Calculate \(2^3\), which is \(8\). Then, negate it to get \(-8\). Add this to \(2\):\[g(2) = -8 + 2 = -6\]

Step 3: Calculate \(g(3)\)

Next, substitute \(x = 3\) into the function \(g(x)\):\[g(3) = -(3)^3 + 3\]

Step 4: Simplify \(g(3)\)

Calculate \(3^3\), which is \(27\). Then, negate it to get \(-27\). Add this to \(3\):\[g(3) = -27 + 3 = -24\]

Key Concepts

Substitution Method

The substitution method is a useful technique for evaluating mathematical expressions. Essentially, it involves replacing variables with given numerical values. In our exercise, the task is to find specific values, such as \(g(2)\) and \(g(3)\), for the given polynomial function. Here's how it works:

• Identify the expression or function where substitution is needed.
• Replace the variable in the expression with the specified value.
• Perform arithmetic calculations to simplify the expression and obtain a result.

When applying this method to the function \(g(x) = -x^3 + x\), we substituted 2 and 3 for \(x\) to find \(g(2)\) and \(g(3)\) respectively. This hands-on approach allows us to clearly see the effects of different inputs on the output of a function by evaluating the function at specified points.

Function Notation

Function notation is a systematic way of representing mathematical functions, providing clarity about which function we are working with and the input variable. It typically takes the form \(f(x)\), but in our case, it's \(g(x)\), which can be read as "\(g\) of \(x\)." This notation conveys:

• The name of the function, which is \(g\) in this instance.
• The variable or input, shown in the parentheses as \(x\).
• The rule or formula describing the function's behavior, such as \(-x^3 + x\).

Using function notation, we specify that we're evaluating the function \(g\) at particular values of \(x\) by writing \(g(2)\) or \(g(3)\). This notation is essential in clearly communicating mathematical ideas and allows us to easily substitute different inputs to understand how the function behaves across different values.

Polynomial Functions

Polynomial functions are algebraic expressions that consist of variables raised to whole-number exponents, combined using addition, subtraction, and multiplication. They can vary in complexity, from simple linear functions to more involved quadratic, cubic, or even higher degree polynomials. The function \(g(x) = -x^3 + x\) is a specific type of polynomial function known as a cubic function because its highest degree term is \(x^3\).

Here's what characterizes polynomial functions:

• They include terms with non-negative integer exponents.
• The coefficients can be real numbers, and sometimes they vary based on the specific function.
• Each term contributes to the overall shape of the graph of the function.

Cubic functions like \(g(x)\) can exhibit interesting features on their graphs such as turning points and changes in concavity. Evaluating such functions at specific points, as in this exercise, helps us understand the output at those particular points and offers insights into the overall behavior of the function.