Problem 1 - Intermediate Algebra

Question

For the functions \(f\) and \(g,\) find \(a .(f+g)(x), b .(f-g)(x)\) \(c .(f \cdot g)(x),\) and \(d .\left(\frac{f}{g}\right)(x) .\) $$f(x)=x-7, g(x)=2 x+1$$

Step-by-Step Solution

Answer

a: \((f+g)(x) = 3x - 6\), b: \((f-g)(x) = -x - 8\), c: \((f \cdot g)(x) = 2x^2 - 13x - 7\), d: \(\left(\frac{f}{g}\right)(x) = \frac{x-7}{2x+1}\)."

Step 1: Compute \((f+g)(x)\)

To find \((f+g)(x)\), we need to add the functions \(f(x)\) and \(g(x)\). So, \((f+g)(x) = (x-7) + (2x+1) = x - 7 + 2x + 1\). Combine like terms to simplify: \(3x - 6\). Thus, \((f+g)(x) = 3x - 6\).

Step 2: Compute \((f-g)(x)\)

To find \((f-g)(x)\), subtract \(g(x)\) from \(f(x)\). So, \((f-g)(x) = (x-7) - (2x+1) = x - 7 - 2x - 1\). Simplify by combining like terms: \(-x - 8\). Thus, \((f-g)(x) = -x - 8\).

Step 3: Compute \((f \cdot g)(x)\)

To find \((f \cdot g)(x)\), multiply \(f(x)\) and \(g(x)\). So, \((f \cdot g)(x) = (x-7)(2x+1)\). Use the distributive property: \((x)(2x) + (x)(1) - (7)(2x) - (7)(1)\) which simplifies to \(2x^2 + x - 14x - 7\). Combine like terms: \(2x^2 - 13x - 7\). Thus, \((f \cdot g)(x) = 2x^2 - 13x - 7\).

Step 4: Compute \(\left(\frac{f}{g}\right)(x)\)

To find \(\left(\frac{f}{g}\right)(x)\), divide \(f(x)\) by \(g(x)\). Write this as: \(\frac{x-7}{2x+1}\). No simplification is possible unless specified by further constraints. Thus, \(\left(\frac{f}{g}\right)(x) = \frac{x-7}{2x+1}\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They do not have an equal sign like an equation, and they represent a value.
In the context of functions, we can express the relationship using expressions. For instance, given functions like \( f(x) = x-7 \) and \( g(x) = 2x+1 \), each function is an algebraic expression. The expression itself can be manipulated using various operations such as addition, subtraction, multiplication, or division to find new expressions.

• Term: A single part of an expression, which can be a number, a variable, or numbers and variables multiplied together. Examples include \( x, 2x \), and \( 7 \).
• Coefficient: The numerical part of a term that is multiplied by the variable. For \( 2x \), the coefficient is 2.
• Constant: A number on its own, like \( -7 \) in the expression \( x-7 \).

Understanding these basic components aids in effectively performing operations on expressions.

Distributive Property

The distributive property is a fundamental rule that applies to expressions involving multiplication over addition or subtraction. It allows us to "distribute" a factor across terms within parentheses.
Let's say you have an expression \( (a+b)(c) \). Here, the distributive property suggests multiplying each term in the parentheses by \( c \):

• \( a \cdot c \)
• \( b \cdot c \)

So, \( (a+b)(c) = ac + bc \).
This property significantly simplifies expressions. In the example given, \( (f \cdot g)(x) = (x-7)(2x+1) \), applying the distributive property helps expand the product to \( 2x^2 + x - 14x - 7 \), which simplifies to \( 2x^2 - 13x - 7 \) by combining like terms.

Function Addition

Function addition involves adding the outputs of two functions. Mathematically, if you have functions \( f(x) \) and \( g(x) \), the sum of these functions is denoted as \( (f + g)(x) \).
To find \( (f+g)(x) \), simply add the two expressions:

• For \( f(x) = x-7 \) and \( g(x) = 2x+1 \),
• \( (f+g)(x) = (x-7) + (2x+1) \)
• Which simplifies to \( 3x - 6 \).

Here, adding the like terms (\( x \) terms with other \( x \) terms and constants with constants) results in a new function, \( 3x - 6 \). Function addition is straightforward and a basic operation with powerful implications in modeling relationships between different quantities.

Function Subtraction

Function subtraction follows the same principle as function addition but involves subtracting one function's output from another's. For two functions \( f(x) \) and \( g(x) \), their difference is expressed as \( (f - g)(x) \).
To compute \( (f-g)(x) \), subtract the expressions:

• Given \( f(x) = x-7 \) and \( g(x) = 2x+1 \),
• \( (f-g)(x) = (x-7) - (2x+1) \)
• This simplifies to \( -x - 8 \).

By combining like terms carefully, the subtraction produces a new function. It's important to keep track of coefficients and negative signs to avoid mistakes in computation.

Function Multiplication

Function multiplication involves finding the product of the outputs of two functions. If \( f(x) \) and \( g(x) \) are your functions, their product is \( (f \cdot g)(x) \).
This can expand to:

• For \( f(x) = x-7 \) and \( g(x) = 2x+1 \),
• \( (f \cdot g)(x) = (x-7)(2x+1) \),
• Use the distributive property to expand and simplify: \( 2x^2 + x - 14x - 7 \).
• Simplifies to \( 2x^2 - 13x - 7 \).

Multiplication of functions can yield quadratics or higher-degree polynomials, providing insights into more complex relationships.

Function Division

Function division represents dividing the output of one function by the output of another, expressed as \( \left(\frac{f}{g}\right)(x) \).
Let's see how this looks for \( f(x) = x-7 \) and \( g(x) = 2x+1 \):

• \( \left(\frac{f}{g}\right)(x) = \frac{x-7}{2x+1} \).

Dividing functions can be more complex, often requiring simplification based on domain constraints or factoring.
In many cases, division doesn't simplify neatly unless there are common factors between numerator and denominator. Paying close attention to what's allowable within the domain of the function is crucial, as division by zero would make the function undefined.