Problem 105 - Precalculus Essentials

Question

Explaining the Concepts: If equations for two functions are given, explain how to obtain the quotient function and its domain.

Step-by-Step Solution

Answer

The quotient function of two given functions, say f(x) and g(x), is obtained by dividing f(x) by g(x), which gives us h(x) = f(x) / g(x). The domain is all the possible values of x that can be input into the function. For a quotient function, it's all real numbers where the denominator, g(x), doesn't equate to zero, as division by zero is undefined.

Step 1: Understand Quotient Function

The quotient of two functions, say f(x) and g(x), is obtained by dividing f(x) by g(x). This gives the quotient function h(x) = f(x) / g(x). This means if you have any input x, you apply f and g to x and then divide the result of f by the result of g.

Step 2: Find the Quotient Function

Now, if f(x) and g(x) are given, the quotient function h(x) = f(x) / g(x) can be directly found by dividing one by the other.

Step 3: Determine Domain of Quotient Function

Remember that mathematically we can't divide by zero. So, we must ensure that the denominator g(x) doesn't equate to zero for any real x. The domain of the quotient function h(x) is all real numbers except where g(x) = 0.

Key Concepts

Functions in Mathematics

A function in mathematics is a relation that uniquely assigns an output to each input from a given set. You can think of it as a mathematical machine: you feed it an input (usually represented as 'x'), and it produces an output (often represented as 'f(x)'). These outputs and inputs are typically numbers, but they can also be other mathematical objects.

For example, if we have a function 'f' that squares its input, the output for the input '3' would be '9', because '3 squared' is '9'. This is written mathematically as 'f(3) = 9'. A key characteristic of a function is that each input is related to exactly one output; there's no ambiguity or multiple outputs for any single input. Functions can be represented in several forms, such as graphs, equations, or tables, each with their own useful applications depending on the scenario.

Domain of a Function

The domain of a function refers to the set of all possible input values (usually 'x') for which the function is defined. It’s essentially the question of 'What can you put into the function?' Not all inputs work for every function, and some could cause problems, like division by zero.

When determining the domain, we look at the limitations imposed by the function's formula. For instance, if a function involves the square root of 'x', then 'x' can't be negative because the square root of a negative number isn’t a real number. Similarly, for a function with a denominator, none of the input values can make the denominator equal zero. Identifying the domain is critical when graphing the function or when combining functions through operations such as addition, subtraction, multiplication, or division.

Division of Functions

When we talk about the division of functions, we’re referring to creating a new function (let's call it 'h') by dividing one function (say 'f') by another (let's say 'g'). This is written as 'h(x) = f(x) / g(x)'. Just like dividing numbers, we cannot divide by zero. Therefore, we need to carefully consider the domain of this new quotient function 'h'.

The domain of 'h' will include all of the input values from the domain of 'f' that do not lead to the denominator ('g(x)') being zero. It's essential to check every point where the denominator 'g(x)' could be zero and exclude those from the domain of ‘h’. This ensures that 'h(x)' is well-defined for all ‘x’ in its domain. In summary, calculating the division of functions helps us understand the behavior of complex relationships between variables by examining how the ratio of their outputs changes with different inputs.