Problem 34 - College Algebra

Question

Find \(f+g, f-g,\) fg, and \(\frac{f}{x}\). Determine the $d o^{2}$$$ f(x)-x-6, g(x)-5 x^{2} $$

Step-by-Step Solution

Answer

\(f(x) + g(x) = 5x^{2} + x - 6\), \(f(x) - g(x) = -5x^{2} + x - 6\), \(f(x) \cdot g(x) = 5x^{2}(x - 6)\), \(\frac{f(x)}{x} = 1 - \frac{6}{x}\)

Step 1: f+g

To perform the operation \(f + g\), simply add the functions \(f(x)\) and \(g(x)\) together:\(f(x) + g(x) = (x - 6) + 5x^{2}\)

Step 2: f-g

For \(f - g\), subtract \(g(x)\) from \(f(x)\):\(f(x) - g(x) = (x - 6) - 5x^{2}\)

Step 3: fg

For the multiplication \(fg\), multiply \(f(x)\) by \(g(x)\):\(f(x) \cdot g(x) = (x - 6) \cdot 5x^{2}\)

Step 4: \(\frac{f}{x}\)

To divide \(f\) by \(x\), divide \(f(x)\) by \(x\):\(\frac{f(x)}{x} = \frac{(x - 6)}{x}\)

Key Concepts

Function Addition

Function addition involves combining two functions by adding their corresponding outputs for any given input. This operation is pretty straightforward and can be done with any two functions, whether they're linear, quadratic, or of another type. For example, given two functions, \( f(x) = x - 6 \) and \( g(x) = 5x^2 \), you simply add the expressions:

• \( f(x) + g(x) = (x - 6) + 5x^2 \).

This results in a new function where each term of the functions is combined by addition. The resulting function is \( f + g = 5x^2 + x - 6 \). Always remember to combine like terms if applicable for efficiency. This new function shows how the sum of two different functions' outputs behaves over their domain.

Function Subtraction

Subtracting functions is a similar process to adding them, but instead, you take one function's output away from the other. Just as with addition, you realign the functions' terms before executing the operation. Using the example functions again, \( f(x) = x - 6 \) and \( g(x) = 5x^2 \), subtraction is as follows:

• \( f(x) - g(x) = (x - 6) - 5x^2 \).

This yields the new function \( f - g = -5x^2 + x - 6 \). It’s crucial to watch your signs here to avoid errors, ensuring each term of \( g(x) \) is subtracted properly. This provides insight into how one function's influence is diminished or enhanced, depending on the operation.

Function Multiplication

Function multiplication involves multiplying the outputs of two functions for a given input. This operation combines the terms of each function for a more complex product. Consider \( f(x) = x - 6 \) and \( g(x) = 5x^2 \). To find their product:

• \( f(x) \cdot g(x) = (x - 6) \cdot 5x^2 \).

The result is achieved by distributing the multiplication across the terms, yielding \( f \cdot g = 5x^3 - 30x^2 \). Multiplying functions highlights the interaction between the elements of each function, potentially leading to a polynomial of higher degree. You’ll notice that a linear function multiplied by a quadratic typically results in a cubic function.

Function Division

Function division splits the output of one function by another. However, division comes with additional complexities, particularly since dividing by a variable can lead to undefined values. Let's divide \( f(x) = x - 6 \) by \( x \):

• \( \frac{f(x)}{x} = \frac{(x - 6)}{x} \).

This operation simplifies to \( 1 - \frac{6}{x} \). It’s important to note that the division by zero is undefined, so the value of \( x \) cannot be zero. Division in functions typically results in a rational function, emphasizing the importance of checking for undefined points in the function's domain. This technique is fundamentally about understanding how one function moderates or scales the output of another function.