Problem 12 - Algebra A Combined Function

Question

If \(P(x)=x^{3}+2 x-3\) and \(Q(x)=7 x+5,\) find each function value. $$ Q(0) $$

Step-by-Step Solution

Answer

The value of the function is 5.

Step 1: Substitute the Given Value into the Function

To find the value of the function \(Q(x)\) at \(x = 0\), simply substitute \(0\) in place of \(x\) in the function \(Q(x) = 7x + 5\).

Step 2: Calculate the Result

Perform the substitution and calculate: \(Q(0) = 7(0) + 5\).

Step 3: Simplify the Expression

Complete the calculation: \(Q(0) = 0 + 5 = 5\).

Key Concepts

Polynomial Functions

Polynomial functions are algebraic expressions that involve sums of powers of a variable, in this case, the variable is often denoted as \(x\). Each term in a polynomial function consists of a coefficient multiplied by a power of the variable. For example, a simple polynomial function can look like this: \(P(x) = x^3 + 2x - 3\).

Key characteristics of polynomial functions include:

• Each term is of the form \(a_i x^i\), where \(a_i\) is a constant coefficient and \(i\) is a non-negative integer exponent.
• The degree of the polynomial is determined by the highest power of \(x\) present.
• Polynomials are continuous and smooth curves when graphed.

Understanding how to evaluate polynomial functions by substituting values into these algebraic expressions is fundamental in solving mathematical problems, just like finding \(Q(0)\) in this case.

Substitution Method

The substitution method is a technique used for evaluating algebraic expressions by replacing variables with numerical values. This method is especially useful in simplifying expressions and solving equations. In the context of functions, substitution involves replacing the variable \(x\) in a function's expression with a specific value.

Here is how you apply the substitution method step-by-step:

• Identify the function you want to evaluate, for example, \(Q(x) = 7x + 5\).
• Substitute the value given, in this case, \(x = 0\), into the function wherever \(x\) appears.
• Solve the resulting expression to find the function’s value at that point.

Using substitution, you make the function easier to handle, as you've seen demonstrated with \(Q(0) = 5\). By replacing \(x\) with \(0\), you quickly performed the necessary arithmetic to obtain the result.

Algebraic Expressions

Algebraic expressions are compositions of variables, numbers, and arithmetic operations (like addition, subtraction, multiplication, etc.). Functions like \(P(x)\) and \(Q(x)\) are common examples of algebraic expressions where each term consists of a product of a constant and a variable raised to some power.

Key facts about algebraic expressions include:

• They can have constants (fixed numbers), variables (letters representing numbers, like \(x\)), and coefficients (numbers multiplying the variables).
• You can simplify them by performing operations, substituting values, or factoring.
• They form the building blocks of more complex mathematical structures like equations and functions.

Understanding and manipulating algebraic expressions is crucial in mathematics as they allow us to represent real-world situations and solve problems like determining function values by substitution.