Problem 3 - Algebra A Combined Function

Question

For the functions \(f\) and \(g\), find a. \((f+g)(x)\), b. \((f-g)(x), c .(f \cdot g)(x)\), and d. \(\left(\frac{f}{g}\right)(x)\). $$ f(x)=x^{2}+1 ; g(x)=5 x $$

Step-by-Step Solution

Answer

a. \(x^2 + 5x + 1\); b. \(x^2 - 5x + 1\); c. \(5x^3 + 5x\); d. \(\frac{x^2 + 1}{5x}\).

Step 1: Understanding the Exercise

We are given two functions, \( f(x) = x^2 + 1 \) and \( g(x) = 5x \). The task is to find the expressions for \((f+g)(x)\), \((f-g)(x)\), \((f \cdot g)(x)\), and \(\left(\frac{f}{g}\right)(x)\).

Step 2: Finding \((f+g)(x)\)

To find \((f+g)(x)\), we add \(f(x)\) and \(g(x)\). Compute: \[(f+g)(x) = f(x) + g(x) = (x^2 + 1) + (5x) = x^2 + 5x + 1.\]

Step 3: Finding \((f-g)(x)\)

For \((f-g)(x)\), subtract \(g(x)\) from \(f(x)\). Compute: \[(f-g)(x) = f(x) - g(x) = (x^2 + 1) - (5x) = x^2 - 5x + 1.\]

Step 4: Finding \((f \cdot g)(x)\)

To calculate \((f \cdot g)(x)\), multiply \(f(x)\) by \(g(x)\). Compute: \[(f \cdot g)(x) = f(x) \cdot g(x) = (x^2 + 1) \cdot (5x) = 5x^3 + 5x.\]

Step 5: Finding \(\left(\frac{f}{g}\right)(x)\)

To find \(\left(\frac{f}{g}\right)(x)\), divide \(f(x)\) by \(g(x)\). Assume \(g(x)\) is non-zero. Compute: \[\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^2 + 1}{5x}.\]

Key Concepts

Function Operations

Functions are like mathematical machines. They take an input, process it, and deliver an output. Function operations involve combining these machines! You can add, subtract, multiply, and even divide functions to form new ones.

When dealing with function operations:

• For addition, \((f + g)(x)\) means you add the outputs of \(f\) and \(g\) for any input \(x\).
• For subtraction, \((f - g)(x)\) means you subtract the output of \(g\) from that of \(f\) for each \(x\).
• For multiplication, \((f \cdot g)(x)\) involves multiplying the outputs of \(f\) and \(g\).
• For division, \(\left(\frac{f}{g}\right)(x)\) means you divide the output of \(f\) by the output of \(g\) given \(g(x) eq 0\).

These operations allow mathematicians to develop new insights and solve more complex problems.

Algebra

Algebra is the language of mathematics where we use symbols and letters to represent numbers and express mathematical relationships. It simplifies complex ideas into smaller expressions that are easier to work with.

When working with function operations in algebra, you'll often:

• Combine like terms. For example, terms like \(x^2 \) and \(5x \) are 'unlike' and can only be combined through operations.
• Apply the distributive property when multiplying. Multiply each term within one function by every term in the other.
• Simplify expressions by adding or subtracting like terms.

Algebra provides the tools needed to manipulate and solve these expressions efficiently.

Polynomials

A polynomial is a specific kind of function composed of variables raised to whole number powers and multiplied by coefficients. They can be simple, like \(x + 2\), or more complex, like \(5x^3 + x^2 + 4x + 7\).

When working with polynomial functions, you often:

• Add and subtract polynomials by combining like terms.
• Multiply polynomials by distributing each term in one polynomial by every term in the other.
• Observe the degree of a polynomial. This is the highest power of the variable in the expression, which affects the shape of its graph.
• Recognize that dividing polynomials can be complex and requires careful attention to zero division issues.

Polynomials are foundational in algebra and appear frequently in both academic problems and real-world applications.

Division by Zero

In mathematics, division by zero is an undefined operation. This means you cannot divide any number by zero. Think of it like trying to distribute zero candies among friends – it’s an impossible task!

When considering function operations, especially in division \(\left(\frac{f}{g}\right)(x)\),

• The denominator \(g(x)\) can't be zero, because dividing by zero has no meaning.
• This means we must check the function \(g(x)\) before performing division to avoid undefined expressions.
• If \(g(x)\) does become zero for a certain \(x\), that value is excluded from the domain (the set of all possible inputs).

By recognizing and accounting for division by zero, we maintain the integrity of mathematical calculations.