Problem 7 - Algebra A Combined Function

Question

If \(P(x)=x^{3}+2 x-3\) and \(Q(x)=7 x+5,\) find each function value. $$ P(2) $$

Step-by-Step Solution

Answer

The function value \(P(2)\) is 9.

Step 1: Identify the function

We are given the function \(P(x) = x^3 + 2x - 3\). Our task is to find \(P(2)\), which means we need to substitute \(x = 2\) into the function \(P(x)\).

Step 2: Substitute x value

Substitute \(x = 2\) into the given function: \(P(2) = (2)^3 + 2(2) - 3\).

Step 3: Calculate the powers

Calculate the power of the term: \((2)^3\) which equals \(8\).

Step 4: Multiply coefficients and add

Calculate \(2 \times 2\) which equals \(4\), and then add these results together: \(8 + 4\).

Step 5: Simplify the expression

The expression becomes \(8 + 4 - 3\). First, add \(8 + 4 = 12\), then subtract 3 to arrive at the final result: \(12 - 3 = 9\).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to whole number powers, multiplied by coefficients. These functions can take on various forms such as linear, quadratic, cubic, and beyond. When dealing with polynomial functions, you're essentially looking at sums of terms, each being a product of a constant coefficient and one or more variables raised to a power.

For example, in the polynomial function given by \( P(x) = x^3 + 2x - 3 \), you observe typical features of a polynomial:

• The term \( x^3 \) is the cubic term because the variable \( x \) is raised to the power of 3.
• The term \( 2x \) is a linear term with a coefficient of 2.
• The constant term, which is just -3, represents a standard number without any variable attached.

Understanding polynomial functions is essential because they model a wide range of natural and practical scenarios, from physics to economics. Their behavior, such as their direction, curvature, and roots, can provide substantial insights into the patterns they represent.

These functions form the backbone of algebra, serving as a bridge to more complex mathematical studies.

Substitution Method

Substitution in mathematics is a technique employed to evaluate functions by replacing the variable of the function with a specific value. This process simplifies working with functions and is particularly useful in evaluating expressions.

In our problem, we need to evaluate \( P(x) = x^3 + 2x - 3 \) at \( x = 2 \). This means we're finding \( P(2) \). To do this, follow these steps:

• Identify the function and the value for substitution (here, \( x = 2 \))
• Replace each occurrence of \( x \) in the function with the number 2
• Calculate the resulting expression which gives us the function value at \( x = 2 \)

After substituting \( x = 2 \) into:\
\[ P(2) = (2)^3 + 2\times2 - 3 \]

Each operation is carried out, leading to the exact value sought for this specific input. Substitution is crucial for evaluating functions and checking equivalences in algebraic manipulations. It is a fundamental skill that enhances understanding of how functions behave with specific inputs.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying mathematical expressions. It's all about using algebraic rules to transform expressions into a more convenient or desired form. This process is used to solve equations, simplify expressions, and evaluate functions.

In our exercise, algebraic manipulation becomes apparent when we simplify the substituted expression.After substitution:
\[ P(2) = 8 + 4 - 3 \]
The manipulation involves several steps:

• First, calculate \( 2^3 \), which is 8, the cube of 2.
• Then, compute the multiplication \( 2 \times 2 = 4 \).
• Finally, sum these results: adding \( 8 + 4 = 12 \).
• Subtract 3 to get the final simplified value, 9.

This example illustrates the use of basic algebraic operations like exponentiation, multiplication, and subtraction, combining to achieve a final result. Mastery of algebraic manipulation not only helps in solving problems but builds the groundwork for solving complex algebraic equations and inequalities.