Problem 30 - College Algebra Essentials

Question

Evaluate each function at the given values of the independent variable and simplify. $$ g(x)=x^{2}-10 x-3 $$ a. \(g(-1)\) b. \(g(x+2)\) c. \(g(-x)\)

Step-by-Step Solution

Answer

The simplified expressions after substituting the specific values into the function \(g(x) = x^{2} -10x -3\) are: a. \(g(-1) = 8\), b. \(g(x+2) = x^{2} -6x -19\), and c. \(g(-x) = x^{2} +10x -3\).

Step 1: Substituting -1 into the Function

Substitute \(x = -1\) into \(g(x) = x^{2} -10x -3\), so that \(g(-1) = (-1)^{2} -10(-1) -3\). This will simplify to \(g(-1) = 1 +10 -3 = 8\).

Step 2: Substituting \(x+2\) into the Function

Substitute \(x = x+2\) into \(g(x) = x^{2} -10x -3\), hence replacing \(x\) with \((x+2)\) gives \(g(x+2) = (x+2)^{2} -10(x+2) -3\). Expanding and simplifying the expression, \(g(x+2) = x^{2} +4x +4 -10x -20 -3 = x^{2} -6x -19\).

Step 3: Substituting \(-x\) into the Function

Substitute \(x = -x\) into \(g(x) = x^{2} -10x -3\), yielding \(g(-x) = (-x)^{2} -10(-x) -3\). Simplifying this expression gives \(g(-x) = x^{2} +10x -3\).

Key Concepts

Substitution in Functions

Substituting values into a function is like replacing a placeholder with a specific target. For example, if you have a function like \(g(x) = x^2 - 10x - 3\), the \(x\) serves as a variable.
Substitution occurs when you replace this \(x\) with a specific number or expression:

• When substituting \(x = -1\), it means wherever you see \(x\), you replace it with \(-1\). So, you calculate \(g(-1)\) by rewriting the function as \((-1)^2 - 10(-1) - 3\).
• This process results in \(g(-1) = 1 + 10 - 3 = 8\).

In essence, substitution allows you to evaluate the function for different values or even other expressions. This is a fundamental skill in algebra that lets you explore how functions behave under various inputs.

Polynomial Functions

Polynomial functions are algebraic expressions that consist of variables and coefficients, combined using only addition, subtraction, and multiplication.
A classic polynomial function is expressed as \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\). In this form, \(a_n, a_{n-1}\), etc., are coefficients, and \(n\) is a non-negative integer representing the highest degree of the polynomial, also known as the order of the polynomial.
Polynomial functions can:

• Have multiple terms like \(x^2 - 10x - 3\) in our example of \(g(x)\).
• Include a constant term, which is the term without any variables, like \(-3\) in \(g(x)\).

These functions are powerful tools in mathematics because they can model a wide range of real-world situations and describe various kinds of relationships.

Simplifying Expressions

Simplifying expressions is an essential step to making algebraic operations more manageable.
It involves combining like terms and performing any arithmetic operations within the expression.
To simplify an expression, follow these steps:

• Combine any like terms. These are terms that have the same variable raised to the same power, such as combining \(-10x\) and \(10x\) in the expression \(g(-x) = x^2 + 10x - 3\).
• Simplify arithmetic within parentheses by distributing any factors outside the parentheses through multiplication, as seen in \(g(x+2) = (x+2)^2 - 10(x+2) - 3\).
• Finally, perform all arithmetic operations to bring the expression to its simplest form, resulting in the cleanest expression possible, like \(x^2 - 6x - 19\) for \(g(x+2)\).

Simplifying isn't just for neatness—it's crucial for making expressions ready for further analysis, easier to interpret, and even prettier on the page!