Problem 40 - 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 \(g(x)\) \(f(x)=10 x-20\) \(g(x)=x-2\)

Step-by-Step Solution

Answer

\((f+g)(x)=11x-22\), \((f-g)(x)=9x-18\), \((f \cdot g)(x)=10x^2-40x+40\), \(\left(\frac{f}{g}\right)(x)=\frac{10x-20}{x-2}\)."

Step 1: Understand the Problem

We need to find the expressions for \((f+g)(x)\), \((f-g)(x)\), \((f \cdot g)(x)\), and \(\left(\frac{f}{g}\right)(x)\)\ using the given functions \(f(x)=10x-20\) and \(g(x)=x-2\).

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

To find \((f+g)(x)\), we add the two functions together: \(f(x) + g(x)\). Substitute the expressions for \(f(x)\) and \(g(x)\):\[(f+g)(x) = (10x - 20) + (x - 2)\]Combine like terms:\[= 11x - 22\]

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

To find \((f-g)(x)\), we subtract \(g(x)\) from \(f(x)\):\[(f-g)(x) = (10x - 20) - (x - 2)\]Distribute the negative sign and combine like terms:\[= 10x - 20 - x + 2\]\[= 9x - 18\]

Step 4: Calculate (f ⋅ g)(x)

To find \((f \cdot g)(x)\), we multiply the functions:\[(f \cdot g)(x) = (10x - 20)(x - 2)\]Apply the distributive property by multiplying each term:\[= 10x(x) - 10x(2) - 20(x) + 20(2)\]\[= 10x^2 - 20x - 20x + 40\]\[= 10x^2 - 40x + 40\]

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

To find \(\left(\frac{f}{g}\right)(x)\), we divide \(f(x)\) by \(g(x)\):\[\left(\frac{f}{g}\right)(x) = \frac{10x - 20}{x - 2}\]This is already in the simplest algebraic form, unless further simplification is possible, which in this case is not because the numerator and denominator do not share common factors that can be cancelled.

Key Concepts

Addition of Functions

In mathematics, adding functions is a way to combine two functions into one. To perform the addition of functions, sum the outputs of each function for a given value of x. For example, if you have two functions, \(f(x)\) and \(g(x)\), the addition of these functions is \((f + g)(x) = f(x) + g(x)\). This results in a new function that you can evaluate for different inputs.

In this specific exercise, where \(f(x) = 10x - 20\) and \(g(x) = x - 2\), you add them like this:

• Combine their expressions: \((f+g)(x) = (10x - 20) + (x - 2)\).
• Combine like terms: \(11x - 22\).

Hence, the resulting function \((f + g)(x) = 11x - 22\) is the function you get from adding \(f(x)\) and \(g(x)\).

This new function can be evaluated in the same manner as any other function to find outputs for specific inputs.

Subtraction of Functions

The subtraction of functions is similar to the addition, but instead, you find the difference between the outputs of two functions. If you have functions \(f(x)\) and \(g(x)\), the result of subtracting \(g(x)\) from \(f(x)\) is \((f - g)(x) = f(x) - g(x)\). This operation helps in finding out how one function represents a change concerning another for the same input.

In our problem, to find \((f-g)(x)\) with \(f(x) = 10x - 20\) and \(g(x) = x - 2\), follow these steps:

• Compute \((f-g)(x) = (10x - 20) - (x - 2)\).
• Distribute the minus sign: \(10x - 20 - x + 2\).
• Simplify to get \(9x - 18\).

Therefore, \((f-g)(x) = 9x - 18\) is the new function derived from the subtraction of \(g(x)\) from \(f(x)\).

This expression shows the variation of one function against another at any given value of x.

Multiplication of Functions

Multiplication of functions involves multiplying together the outputs of two functions for each specific input value. Using the notation, \((f \cdot g)(x) = f(x) \cdot g(x)\), the result is a new function. This operation can be useful in contexts where you want to see how the multiplication of quantities varies with each input.

For \(f(x) = 10x - 20\) and \(g(x) = x - 2\):

• Calculate \((f \cdot g)(x) = (10x - 20) \cdot (x - 2)\).
• Use the distributive property to multiply each term: \(= 10x(x) - 10x(2) - 20(x) + 20(2)\).
• Simplify: \(10x^2 - 40x + 40\).

This yields \((f \cdot g)(x) = 10x^2 - 40x + 40\), a quadratic expression that results from multiplying the given functions.

Quadratic functions exhibit parabolic behavior, meaning they curve upwards or downwards based on the leading coefficient's sign.

Division of Functions

Dividing functions involves performing a division operation between their outputs. The basic format is given by \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\). It is important to consider restrictions where the denominator cannot be zero, as division by zero is undefined.

Given \(f(x) = 10x - 20\) and \(g(x) = x - 2\), you can find the division of these functions as follows:

• Set up the division: \(\left(\frac{f}{g}\right)(x) = \frac{10x - 20}{x - 2}\).

The current division \(\left(\frac{10x - 20}{x - 2}\right)\) is the simplest form unless the numerator and denominator have common factors to simplify, which is not apparent here.

Remember that in division operations, it’s crucial to identify and note any values of x for which \(g(x)\) becomes zero to avoid undefined expressions. In this case, \(x = 2\) would make the denominator zero, so it's excluded from the domain.