Problem 57 - Algebra 2

Question

Find \((f+g)(x),(f-g)(x),(f \cdot g)(x),\) and \(\left(\frac{f}{g}\right)(x)\) for each \(f(x)\) and \(g(x)\) $$ \begin{array}{l}{f(x)=10 x-20} \\ {g(x)=x-2}\end{array} $$

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: Adding Functions

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

Step 2: Subtracting Functions

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

Step 3: Multiplying Functions

To find \((f \cdot g)(x)\), multiply the functions \(f(x)\) and \(g(x)\):\[(f \cdot g)(x) = f(x) \cdot g(x) = (10x - 20)(x - 2)\]Use the distributive property (FOIL method) to expand:\[f(x) \cdot g(x) = (10x \cdot x) + (10x \cdot (-2)) - (20 \cdot x) - (20 \cdot (-2))\]\[= 10x^2 - 20x - 20x + 40 = 10x^2 - 40x + 40\]

Step 4: Dividing Functions

To find \(\left(\frac{f}{g}\right)(x)\), divide \(f(x)\) by \(g(x)\):\[\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{10x - 20}{x - 2}\]Check for any restrictions: The denominator \(x - 2\) cannot be zero, thus \(x eq 2\). The division is simplified as it stands because no common factors exist.

Key Concepts

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers and coefficients. They can have multiple terms, with each term being a product of a constant and a variable raised to a power. A simple polynomial example is \(f(x) = 10x - 20\), which is a linear polynomial. This means it is a polynomial of degree one, as the highest power of \(x\) is one.
Polynomials are named according to their degree:

• Constant polynomials, where the degree is zero (e.g., \(f(x) = 3\)).
• Linear polynomials, with the highest degree being one.
• Quadratic polynomials, which have a degree of two.
• Cubic polynomials, with a degree of three, and so on.

Understanding polynomial functions is crucial as they form the basis for many areas of math and science.

Addition of Functions

In order to add functions, simply add their corresponding outputs for any given input. This is analogous to adding two numbers to get a sum.
For given functions \(f(x) = 10x - 20\) and \(g(x) = x - 2\), we calculate \((f+g)(x)\) as follows:

• Add the like terms together: \((f+g)(x) = f(x) + g(x) = (10x - 20) + (x - 2)\)
• Simplify by combining like terms: \(11x - 22\)

This resulting function \((f+g)(x)\) represents the sum of the two original functions. It's important to line up and combine terms to ensure accuracy.

Subtraction of Functions

Subtracting functions is similar to addition, except you subtract the output of one function from the output of another for the same input. With \(f(x) = 10x - 20\) and \(g(x) = x - 2\), we find \((f-g)(x)\):

• Subtract the function \(g(x)\) from \(f(x)\): \((f-g)(x) = f(x) - g(x) = (10x - 20) - (x - 2)\)
• Change the signs of the terms in \(g(x)\): \(10x - 20 - x + 2\)
• Combine like terms: \(9x - 18\)

This gives us the new function \((f-g)(x)\), which shows the difference between the two functions. Always be careful to distribute negative signs appropriately.

Multiplication of Functions

Multiplying functions involves finding the product of their respective outputs for each input. When multiplying polynomials, it's important to use the distributive property properly. For \(f(x) = 10x - 20\) and \(g(x) = x - 2\):

• Distribute each term in \(f(x)\) over every term in \(g(x)\):
• \((f \cdot g)(x) = (10x \cdot x) + (10x \cdot (-2)) - (20 \cdot x) - (20 \cdot (-2))\)
• Result in \(10x^2 -20x -20x + 40\)
• Simplify to get \(10x^2 - 40x + 40\)

This process results in a new polynomial which represents the product of the two functions.

Division of Functions

Dividing one function by another results in a new function that represents the quotient of their outputs. In the case of \(f(x) = 10x - 20\) and \(g(x) = x - 2\), the operation \(\left(\frac{f}{g}\right)(x)\) is:

• \(\left(\frac{f}{g}\right)(x) = \frac{10x - 20}{x - 2}\)
• Ensure \(g(x) eq 0\): here, \(x - 2 eq 0\), so \(x eq 2\)
• This is simplified to the expression shown since there are no common factors to cancel.

Division of polynomial functions involves identifying any restrictions on \(x\) where the denominator becomes zero. Thus, it's essential to address these restrictions to avoid undefined expressions.