Problem 41 - Algebra 2

Question

Find \((f+g)(x),(f-g)(x),(f \cdot g)(x),\) and \(\left(\frac{f}{g}\right)\) for each \(f(x)\) and \(g(x)\) \(f(x)=4 x^{2}-9\) \(g(x)=\frac{1}{2 x+3}\)

Step-by-Step Solution

Answer

Solutions are expressions combining \(f(x)\) and \(g(x)\).

Step 1: Understand the Functions Provided

You are given two functions, \(f(x)=4x^2-9\) and \(g(x)=\frac{1}{2x+3}\). You need to determine the sum, difference, product, and quotient of these functions.

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

The sum of the functions \(f\) and \(g\) is given by \((f+g)(x) = f(x) + g(x)\). Substitute the given expressions to get: \( (f+g)(x) = (4x^2 - 9) + \frac{1}{2x+3} \).

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

The difference of the functions \(f\) and \(g\) is given by \((f-g)(x) = f(x) - g(x)\). Substitute the provided expressions to get: \((f-g)(x) = (4x^2 - 9) - \frac{1}{2x+3} \).

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

The product of the functions \(f\) and \(g\) is calculated by multiplying them together: \((f \cdot g)(x) = f(x) \cdot g(x)\). Use the given functions to get: \((f \cdot g)(x) = (4x^2 - 9) \cdot \frac{1}{2x+3} \).

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

The quotient of the functions \(f\) and \(g\) can be found by dividing \(f(x)\) by \(g(x)\): \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\). Substitute the given expressions to get: \(\left(\frac{f}{g}\right)(x) = \frac{4x^2 - 9}{\frac{1}{2x+3}} = (4x^2 - 9) \cdot (2x+3)\).

Key Concepts

Sum of Functions

To find the sum of two functions, we simply add their expressions together. Consider the functions given:

• \( f(x) = 4x^2 - 9 \)
• \( g(x) = \frac{1}{2x+3} \)

The sum, denoted as \((f+g)(x)\), involves adding these two function expressions:\[(f+g)(x) = (4x^2 - 9) + \frac{1}{2x+3}\]Adding functions is straightforward, but remember:

• Ensure to have a common denominator if fractions are involved.
• Combining terms is simple, but be aware of like terms.

In this case, simplifying further may involve finding a common denominator and requiring more algebraic manipulation.

Difference of Functions

Calculating the difference between functions follows a similar principle to evaluating their sum, but with subtraction. Using the same functions:

• \( f(x) = 4x^2 - 9 \)
• \( g(x) = \frac{1}{2x+3} \)

The difference is represented as \((f-g)(x)\), calculated by:\[(f-g)(x) = (4x^2 - 9) - \frac{1}{2x+3}\]Points to keep in mind:

• Simplify the expression ensuring the operations on all terms are done correctly.
• Watch out for subtraction across fractions, as it may need a common denominator.

Again, while we start with expressions, further simplification relies on careful algebra with fractions.

Product of Functions

When multiplying two functions, the operation entails distributing one expression across the other. Let us take a look at the functions given:

• \( f(x) = 4x^2 - 9 \)
• \( g(x) = \frac{1}{2x+3} \)

The product of these functions is described as \((f \cdot g)(x)\), found by:\[(f \cdot g)(x) = (4x^2 - 9) \cdot \frac{1}{2x+3}\]For a step-by-step multiplication process:

• Distribute each term in the polynomial to the rational function.
• Simplify by combining like terms, if possible.

The outcome of this product may be complex, and simplify cautiously is needed to avoid algebraic errors.

Quotient of Functions

The quotient of functions involves dividing one function by another, which is different from the previous operations due to involved rational expressions. With the same functions:

• \( f(x) = 4x^2 - 9 \)
• \( g(x) = \frac{1}{2x+3} \)

The quotient is expressed as \(\left(\frac{f}{g}\right)(x)\), calculated using:\[ \left(\frac{f}{g}\right)(x) = \frac{4x^2 - 9}{\frac{1}{2x+3}}\]Multiplying by the reciprocal:\[= (4x^2 - 9) \cdot (2x + 3) \]Here is what to consider:

• Convert the division into multiplication by using the reciprocal of \(g(x)\).
• Be cautious of restrictions on the domain due to potential divisions by zero.

The result will often require simplification, paying special attention to potential reductions or factoring opportunities.