Problem 29 - College Algebra Essentials

Question

Evaluate each function at the given values of the independent variable and simplify. $$ g(x)=x^{2}+2 x+3 $$ a. \(g(-1)\) b. \(g(x+5)\) c. \(g(-x)\)

Step-by-Step Solution

Answer

a) g(-1) = 2, b) g(x+5) = \(x^2 + 12x + 28\), c) g(-x) = \(x^2 - 2x + 3\)

Step 1: Evaluating g(-1)

To evaluate the function g at -1, substitute -1 for x in the function: \[g(-1) = (-1)^2 + 2(-1) + 3\] Simplify this expression using order of operations: square first, followed by multiplication and addition. This yields a result of 2.

Step 2: Evaluating g(x+5)

To evaluate the function g at x+5, substitute x+5 for x in the function: \[g(x+5) = (x+5)^2 + 2(x+5) + 3\] Simplify this expression by expanding the square and combining like terms. The final simplification results in \[x^2 + 12x + 28\].

Step 3: Evaluating g(-x)

To evaluate the function g at -x, substitute -x for x in the function: \[g(-x) = (-x)^2 + 2(-x) + 3\] Simplify this expression by squaring -x to get x^2, reversing the sign on the 2x term, and combining like terms, yields \[x^2 - 2x + 3\]

Key Concepts

Order of Operations

Understanding the order of operations is key when evaluating any mathematical expression. This concept ensures calculations are performed correctly, avoiding any mix-ups in results. In mathematics, the order of operations is a set of rules used to clarify which procedures to perform first when evaluating expressions.

Here's the usual order one should follow, often remembered by the acronym PEMDAS:

• Parentheses - Calculate expressions inside parentheses first.
• Exponents - Next, perform calculations involving exponents or powers.
• Multiplication and Division - Then, from left to right, take care of multiplication and division.
• Addition and Subtraction - Lastly, execute addition and subtraction, also from left to right.

In the given exercise, when evaluating functions like \(g(-1)\), it is crucial to perform \((-1)^2\) before the other operations to get the correct result.

Polynomial Functions

Polynomial functions, like \(g(x) = x^2 + 2x + 3\), are algebraic expressions that consist of variables and coefficients. They involve terms of varying degrees, where each term represents a variable, raised to an exponent and multiplied by a coefficient.

Key features of polynomial functions include:

• Degree: The highest power of the variable in the polynomial determines its degree. For \(g(x)\), the degree is 2, because \(x^2\) is the highest exponent in the expression.
• Standard Form: A polynomial is in standard form when its terms are ordered from highest to lowest power of the variable.
• Coefficients: These are the numerical factors multiplied by each variable term, such as 1, 2, and 3 in our example.

Understanding polynomial functions helps in evaluating them accurately by identifying and correctly performing operations based on each term's degree and coefficients.

Substitution Method

The substitution method is a technique used in algebra to find specific values of a function by replacing the variable with a given number or expression. This method simplifies the function into a simpler arithmetic or algebraic expression.

In the original problem, we see examples of substitution applied:

• For \(g(-1)\), substitute -1 for \(x\) and simplify the expression using the order of operations.
• For \(g(x+5)\), replace \(x\) with \(x+5\) and simplify to find the new polynomial form.
• For \(g(-x)\), substitute -x for \(x\) and ensure the expression reflects the changed variable.

Substitution is essential in evaluating and solving algebraic problems, providing a powerful method to understand the behavior of functions under different inputs.