Problem 46 - Elementary Algebra

Question

Given \(f(x)=x+13 x\) and \(g(x)=2 x-8\), calculate \((f-g)(x)\) and state the restrictions.

Step-by-Step Solution

Answer

\((f-g)(x) = 12x + 8\) with no restrictions.

Step 1: Simplify the Expression for f(x)

First, simplify the given function \( f(x) = x + 13x \). Combine like terms to get the simplified expression \( f(x) = 14x \).

Step 2: Rearrange the Expression for g(x)

The expression for \( g(x) \) is already simplified and given by \( g(x) = 2x - 8 \).

Step 3: Compute (f-g)(x)

To find \( (f-g)(x) \), subtract the expression for \( g(x) \) from the expression for \( f(x) \). This yields: \( (f-g)(x) = f(x) - g(x) = 14x - (2x - 8) \).

Step 4: Distribute and Simplify

Distribute the negative sign across the terms in \( g(x) \) to remove the parentheses: \( (f-g)(x) = 14x - 2x + 8 \). Combine the like terms: \( (f-g)(x) = (14x - 2x) + 8 = 12x + 8 \).

Step 5: Identify Restrictions on x

Both \( f(x) \) and \( g(x) \) are polynomials, meaning \( x \) can be any real number. Thus, there are no restrictions on \( x \).

Key Concepts

Function Subtraction

Function subtraction is an operation where you subtract one function from another, similar to how you might subtract numbers or simple algebraic expressions. If you have two functions, say \( h(x) \) and \( k(x) \), the difference is expressed as \((h-k)(x) = h(x) - k(x)\). This notation tells us to subtract the entire expression of \( k(x) \) from \( h(x) \).

This subtraction involves distributing a negative sign across the entirety of the function you're subtracting. It's important to keep track of this sign to avoid making errors in simplifying the expression. For example, when subtracting \( g(x) = 2x - 8 \) from \( f(x) = 14x \), be sure to distribute the negative sign to get \( 14x - 2x + 8 \).

Function subtraction is powerful because it can be applied to derive new functions, combine functions to simplify complex problems, or compare the behavior of different functions over a set of values.

Algebraic Expressions

Algebraic expressions are mathematical phrases that involve numbers, variables, and operational symbols, such as addition, subtraction, multiplication, and division. A polynomial, like \( f(x) = 14x \), is a specific type of algebraic expression composed solely of terms that take the form \( ax^n \), where \( a \) is a coefficient, and \( n \) is a non-negative integer.

In our example, both \( f(x) = 14x \) and \( g(x) = 2x - 8 \) are algebraic expressions. These involve real numbers and the same variable, \( x \). The first step often involves simplifying these expressions by combining like terms. For instance, initially, \( f(x) = x + 13x \) is simplified to \( f(x) = 14x \) by combining \( x \) terms.

• Terms: Individual components of expressions separated by addition or subtraction.
• Like Terms: Terms that share the same variable and exponent, which can be combined.
• Coefficient: The numerical factor in a term that multiplies the variable.

These simplifications can help reduce the complexity and afford clearer insight into the relationships and operations required between functions.

Real Numbers

Real numbers are a vast and critical component of mathematics, encompassing all the numbers you are most likely familiar with. They include integers, fractions, and irrational numbers — essentially, any value that can represent a distance along a number line.

When working with functions like \( f(x) \) and \( g(x) \) from algebra, the variable \( x \) often represents any real number. This means \( x \) could take on an infinite number of possible values, all lying somewhere on the number line. In the context of polynomial functions (like those we are working with), there are generally no restrictions on \( x \) regarding the domain of real numbers. The function remains valid for any \( x \) you choose.

• Integers: Whole numbers that can be positive, negative, or zero.
• Rational Numbers: Numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \).
• Irrational Numbers: Numbers that cannot be precisely expressed as simple fractions, such as \( \pi \) or \( \sqrt{2} \).

Understanding that these numbers form the basis for polynomial variables helps in grasping why no restrictions apply to \( x \) within this problem. It's the real number system that allows for the infinite domain typically associated with polynomial operations.