Problem 84 - Algebra 2

Question

Find the domain and the range of each function. $$ y=\log (x-t) $$

Step-by-Step Solution

Answer

The domain of the function \(y=\log(x-t)\) is all values of x greater than t (i.e. \(x > t\)). The range is all real numbers.

Step 1: Find the Domain

A log function is only defined if the argument inside the parenthesis, \(x-t\), is greater than zero, because logarithm function is not defined for negative numbers and zero. So, you need to solve this inequality for x. \[x-t > 0\] After solving, you get \(x > t\). So, the domain of the function \(y=\log (x-t)\) is all values of x greater than t (i.e. \(x > t\)).

Step 2: Find the Range

The range of a logarithmic function is all real numbers, because a logarithmic function can take any real value. No matter what the base or the argument of a logarithm is, the output can be any real number. So, the range of the function \(y=\log (x-t)\) is all real numbers.

Key Concepts

Domain of a Function

In mathematics, the **domain of a function** is a critical concept, especially when working with functions such as logarithms. The domain is simply the set of all possible input values (usually represented by *x*) that allow the function to produce a valid output.

When dealing with logarithmic functions, understanding their domain is vital because logarithms have specific restrictions.

• Logarithmic functions, like \(y=\log(x-t)\), require the argument inside the logarithm, \(x-t\), to be greater than zero.
• This is because the logarithm of a non-positive number (i.e., zero or a negative number) is undefined in the realm of real numbers.

As a solution for the exercise, solving the inequality \(x-t>0\), we find that the domain is \(x > t\).

This tells us that the function can only take values greater than *t*. If *t* were 3, for instance, then \(x\) must be greater than 3 for \(y = \log(x-3)\) to output a valid number.

Range of a Function

The **range of a function** is all about understanding what kinds of output values a function can provide. While the domain tells us about possible inputs, the range informs us about possible outputs.

For logarithmic functions, such as \(y = \log(x-t)\), the situation is quite straightforward.

• The range of a logarithmic function is the entire set of real numbers.
• This means that no matter what real number you provide, there is always a way to "reach" it using a logarithmic function.
• For any given real number \(y\) as the output, you can find some \(x\) that makes \(\log(x-t)\) equal \(y\).

So, the range of \(y = \log(x-t)\) is \(-\infty < y < \infty\).

Unlike polynomial or exponential functions, which might have more restricted ranges, logarithmic functions shine in their ability to cover the full spectrum of real numbers.

Inequalities

Working with **inequalities** is essential in various areas of math, including when dealing with domains of functions. Inequalities can help us determine where a function is defined.

In the context of a logarithmic function like \(y = \log(x-t)\), understanding inequalities helps in defining its domain.

• We start with the expression \(x-t > 0\), which arises from the requirement that the argument of a log function must be positive.
• This inequality represents all the values of \(x\) that keep \(x-t\) positive.

Solving \(x-t > 0\) is simple: You add \(t\) to both sides, leading to \(x > t\).

This application of inequalities to functions demonstrates a key technique for defining valid input sets. Similarly, inequalities are crucial in many other areas of math, helping to express limits and possibilities in algebraic structures.