Problem 19 - 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 \(g(x)\) $$ \begin{array}{l}{f(x)=x^{2}-x-6} \\ {g(x)=\frac{x-3}{x+2}}\end{array} $$

Step-by-Step Solution

Answer

The combined functions are: - \((f+g)(x) = x^2 - x - 6 + \frac{x-3}{x+2}\) - \((f-g)(x) = x^2 - x - 6 - \frac{x-3}{x+2}\) - \((f \cdot g)(x) = (x^2 - x - 6)(\frac{x-3}{x+2})\) - \(\left(\frac{f}{g}\right)(x) = \frac{(x^2 - x - 6) \cdot (x+2)}{x-3}\)

Step 1: Add the Functions

To find \[(f+g)(x) = f(x) + g(x)\], simply add the given expressions for \(f(x)\) and \(g(x)\):\[f(x) = x^2 - x - 6\]and\[g(x) = \frac{x-3}{x+2}\]Result:\[(f+g)(x) = \left(x^2 - x - 6\right) + \left(\frac{x-3}{x+2}\right)\]

Step 2: Subtract the Functions

To find \[(f-g)(x) = f(x) - g(x)\], subtract \(g(x)\) from \(f(x)\):\[f(x) = x^2 - x - 6\]and\[g(x) = \frac{x-3}{x+2}\]Result:\[(f-g)(x) = \left(x^2 - x - 6\right) - \left(\frac{x-3}{x+2}\right)\]

Step 3: Multiply the Functions

To find \[(f \cdot g)(x) = f(x) \cdot g(x)\], multiply \(f(x)\) by \(g(x)\):\[f(x) = x^2 - x - 6\]and\[g(x) = \frac{x-3}{x+2}\]Result:\[(f \cdot g)(x) = \left(x^2 - x - 6\right) \cdot \left(\frac{x-3}{x+2}\right)\]

Step 4: Divide the Functions

To find \[\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\], divide \(f(x)\) by \(g(x)\):\[f(x) = x^2 - x - 6\]and\[g(x) = \frac{x-3}{x+2}\]Result:\[\left(\frac{f}{g}\right)(x) = \frac{x^2 - x - 6}{\frac{x-3}{x+2}}\]Simplifying the expression yields:\[\left(\frac{f}{g}\right)(x) = \frac{(x^2 - x - 6) \cdot (x+2)}{x-3}\]

Key Concepts

Addition of Functions

When we add two functions, we are essentially combining their outputs for each input value. Given two functions, say \(f(x)\) and \(g(x)\), the addition of these functions is denoted as \(\ (f+g)(x) = f(x) + g(x)\).
For the functions \(f(x) = x^2 - x - 6\) and \(g(x) = \frac{x-3}{x+2}\), the addition process involves:

• Writing the expression for each function.
• Add them mathematically to form a new expression.

So, \((f+g)(x)\) becomes \((x^2 - x - 6) + \left(\frac{x-3}{x+2}\right)\). This does not simplify further due to the presence of a radical in \(g(x)\).
Always remember that to add functions with different denominators involves further algebraic manipulation, which is necessary for simplification when applicable.

Subtraction of Functions

Subtracting functions is similar to adding them, but instead, we find the difference between the two functions' outputs. This is represented as \((f-g)(x) = f(x) - g(x)\). Let's use the same functions \(f(x) = x^2 - x - 6\) and \(g(x) = \frac{x-3}{x+2}\). The subtraction process looks like this:

• Express \(f(x)\) and \(g(x)\).
• Subtract \(g(x)\) from \(f(x)\).

Our result is \((f-g)(x) = (x^2 - x - 6) - \left(\frac{x-3}{x+2}\right)\). Again, simplification depends on algebraic manipulation, often requiring the establishment of a common denominator if the functions have rational expressions. Make sure you distribute the negative sign across the subtracted function to avoid errors.

Multiplication of Functions

Multiplying functions involves creating a product of their individual outputs. The function multiplication is denoted by \((f \cdot g)(x) = f(x) \cdot g(x)\). With our functions \(f(x) = x^2 - x - 6\) and \(g(x) = \frac{x-3}{x+2}\), follow these steps to multiply:

• Multiply each component of \(f(x)\) by the entire expression of \(g(x)\).
• Simplify the resulting product if possible.

Here, \((f \cdot g)(x) = (x^2 - x - 6) \cdot \left(\frac{x-3}{x+2}\right)\). This multiplication results in a compacted expression, but simplification is key, particularly focusing on factoring and canceling any common factors if possible.

Division of Functions

Division of functions requires dividing the output of one function by the other, expressed as \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\). For \(f(x) = x^2 - x - 6\) and \(g(x) = \frac{x-3}{x+2}\), here is how you approach the division:

• Write out the fraction with \(f(x)\) as the numerator and \(g(x)\) as the denominator.
• Multiply by the reciprocal of \(g(x)\) to simplify.

The result becomes \(\left(\frac{f}{g}\right)(x) = \frac{x^2 - x - 6}{\frac{x-3}{x+2}}\). To simplify, multiply the numerator by the reciprocal of the denominator, resulting in \(\frac{(x^2 - x - 6) \cdot (x+2)}{x-3}\). Simplification checks could involve factoring terms in the numerator or looking for cancelable terms to make the expression clearer.