Problem 19 - Algebra 2

Question

Let \(f(x)=9 x\) and \(g(x)=3 x .\) Find \((f \cdot g)(x)\) and \(\left(\frac{f}{g}\right)(x)\) and their domains.

Step-by-Step Solution

Answer

\((f \cdot g)(x) = 27x^2\) with domain all real numbers, and \(\left(\frac{f}{g}\right)(x) = 3\) with domain all real numbers except \(0\).

Step 1: Find the Product of the Functions

To find \((f \cdot g)(x)\), multiply the two functions \(f(x) = 9x\) and \(g(x)=3x\). That is \((f \cdot g)(x) = f(x) \cdot g(x) = 9x \cdot 3x = 27x^2\). So, \((f \cdot g)(x) = 27x^2\). The domain of \((f \cdot g)(x)\) is all real numbers.

Step 2: Find the Quotient of the Functions

To find \(\left(\frac{f}{g}\right)(x)\), divide the two functions \(f(x) = 9x\) by \(g(x)=3x\). That is, \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{9x}{3x}\). This simplifies to \(\left(\frac{f}{g}\right)(x) = 3\). The domain of \(\left(\frac{f}{g}\right)(x)\) is all real numbers except \(0\), since \(0\) would make the denominator equals to \(0\), which is undefined in the division operation.

Key Concepts

Function multiplication

When we talk about function multiplication, we're referring to a process where we multiply two functions together. In this example, the functions given are \(f(x) = 9x\) and \(g(x) = 3x\). To find the product \((f \cdot g)(x)\), we simply multiply the outputs of these functions.

• Start by multiplying each part: \( f(x) \cdot g(x) = 9x \cdot 3x \).
• This multiplication results in \( 27x^2 \). The "9" multiplies with the "3", and the "x" multiplies with "x", giving \( x^2 \).

The exciting thing about function multiplication is that it follows the same rules as algebraic multiplication. This means you can multiply each term as you would with numbers, just with an added twist of using functions.In terms of the domain, the domain for \((f \cdot g)(x) = 27x^2\) is all real numbers. That’s because there's no division involved or square roots, which sometimes restrict the domain. So, any real number is a valid input.

Function division

Function division works a bit like how you would divide numbers—except here, we’re dividing function outputs. Starting with the given functions \( f(x) = 9x \) and \( g(x) = 3x \), dividing these functions is straightforward.

• To divide \(f(x)\) by \(g(x)\), use the fraction form: \(\frac{f(x)}{g(x)} = \frac{9x}{3x} \).
• Cancel out the "x" from the numerator and denominator (as long as \(x eq 0\)), so you’re left with \(3\).

Thus, \(\left( \frac{f}{g} \right)(x) = 3\). Function division simplifies the problem by reducing complexity through cancellation, which is both satisfying and useful.Regarding the domain, for \(\left( \frac{f}{g} \right)(x) = 3\), we need to exclude \(x = 0\). Inputting \(0\) into the function will result in division by zero, which isn't permissible in mathematics. Therefore, all real numbers except \(0\) form the domain.

Domain of a function

Understanding the concept of the domain of a function is crucial. The domain is essentially all the possible inputs (or x-values) that a function can accept without breaking mathematical rules.For function multiplication, like \((f \cdot g)(x) = 27x^2\), the domain is all real numbers. Why? Because you can multiply real numbers without restriction; no fractions or roots are involved which might limit inputs.With function division, say for \(\left( \frac{f}{g} \right)(x) = 3\), the concept of the domain gets a bit more complex because we have to consider when the denominator could be zero. Dividing by zero is undefined, so \(x = 0\) cannot be a part of the domain. This means the domain must be all real numbers except where the denominator becomes zero.

• This careful consideration is at the heart of understanding domains in mathematical functions, especially in operations like division.

To properly analyze a function’s domain, always think about the "rules" that come with operations—such as not dividing by zero or taking square roots of negative numbers. Doing so ensures solid understanding and prevents mathematical errors.