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Chapter 8: Problem 77

Find the domain and the range of each function. $$ y=3 \log x $$

Short Answer

Expert verified
The domain of the function \( y = 3 \log x \) is \( (0, \infty) \) and the range is \( (-\infty, \infty) \).

Step by step solution

01

Finding the domain

Like we mentioned in the analysis, 'x' inside the logarithm has to be greater than zero. So in this function \(\ y = 3 log x \) , the domain is \( x > 0 \). All values of x that are allowed by this inequality constitute the domain of the function.
02

Finding the range

In a logarithmic function, the resulting y-value can be any real number, depending on the value of 'x'. Hence, the range of this function is all real numbers ( \( -\infty < y < \infty \) ).
03

Writing the domain and range

Now, we write the domain and range of the function in interval notation. The domain, \( x > 0 \), is represented as \( (0, \infty) \). The range, which is all real numbers, is represented as \( (-\infty, \infty) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain of a Function
The domain of a function refers to all the possible input values (usually denoted as 'x') which the function can accept without resulting in any undefined or erroneous expressions. For the function \( y = 3 \log x \), the expression inside the logarithm, which is 'x', must be greater than zero. This is because the logarithm of zero or a negative number is undefined. Therefore, the domain of \( y = 3 \log x \) is all positive real numbers. We represent this in interval notation as \( (0, \infty) \), meaning x can be any number greater than zero.

To summarize, when determining the domain for a logarithmic function:
  • Ensure that the argument of the logarithm (what you're taking the log of) is greater than zero.
  • For \( \log(x) \), the condition is \( x > 0 \).
  • Express this condition in interval notation to describe the domain comprehensively.
Range of a Function
The range of a function consists of all the possible output values (denoted as 'y') that a function can produce. For a logarithmic function like \( y = 3 \log x \), it's important to understand that the outputs can span all real numbers. This is because, as 'x' increases from values just above zero to infinity, 'y' will increase continuously without bound. Additionally, as 'x' approaches zero from the right side, 'y' decreases towards negative infinity.

In interval notation, we express this entire spectrum of y-values as \( (-\infty, \infty) \). This notation tells us that any real number could be an output of the function, indicating a wide and unbounded range.
  • In general, logarithmic functions have inverses that are exponential, contributing to why their range covers all real numbers.
  • It is important to realize that the range shows how a function behaves in terms of outputs, encompassing every result you could expect as inputs change.
Interval Notation
When describing the domain and range of functions, interval notation provides a concise and clear way to express the set of numbers involved. Interval notation involves writing a pair of numbers which represent the start and end of an interval. For instance:
  • Round brackets \( () \) indicate that an endpoint is not included (open interval).
  • Square brackets \( [] \) indicate that an endpoint is included (closed interval).

In the expression for the domain of \( y = 3 \log x \) as \( (0, \infty) \):
  • The round bracket next to 0 means that 0 is not included, reflecting that x must be greater than, not equal to, zero.
  • The infinity symbol \( \infty \) always uses a round bracket because infinity isn't a number we can reach or include.

For the range, \( (-\infty, \infty) \) implies that 'y' can truly be any real number with no bounds.

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