Problem 83 - Algebra 2

Question

Find the domain and the range of each function. $$ y=\log _{2} x+\frac{1}{3} $$

Step-by-Step Solution

Answer

The domain of the function \(y = \log_{2}{x} + \frac{1}{3}\) is all real numbers x such that x > 0. The range of the function is all real numbers.

Step 1: Finding the domain

The domain of a logarithmic function is where the function is defined, so it's the set of all real number x such that x > 0. This is due to the fact that we cannot take the logarithm of a negative number or zero. So, for the function \(y = \log_{2}{x} + \frac{1}{3}\), the domain will be all real numbers x such that x > 0.

Step 2: Finding the range

The range of a logarithm function is all real numbers. The function is shifted \(\frac{1}{3}\) units upward, but this shift does not change the range of the function. Therefore, the range of \(y = \log_{2}{x} + \frac{1}{3}\) is all real numbers.

Key Concepts

Domain of a Function

When dealing with mathematical functions, the domain is one of the fundamental concepts to grasp. Simply put, the domain of a function is the set of all possible input values (or "x" values) that make the function work without any issues. In the context of logarithmic functions, understanding the domain is crucial because logarithms have specific requirements.

Logarithmic functions are only defined for positive input values. This means a logarithmic function like \( y = \log_{2}{x} + \frac{1}{3} \) is only defined when \( x > 0 \). Here's why:

• No Negative Inputs: You cannot take the logarithm of a negative number.
• No Zero: The logarithm of zero is undefined.

The rule of thumb is that the base (in this case, 2) raised to any real number cannot result in a negative number or zero. Thus, for safe calculations and to keep within the defined limits of the function, \( x \) should always be greater than zero. This gives us the domain of the function: all real numbers \( x \) where \( x > 0 \).

Understanding the domain is vital because it tells you where the function lives and operates without running into mathematical errors.

Range of a Function

The range of a function is all the possible output values (or "y" values) that you can get from plugging various inputs into the function. For logarithmic functions, which inherently grow very slowly, understanding the range provides insight into their potential behaviors.

The function \( y = \log_{2}{x} + \frac{1}{3} \) has its logarithmic component, \( \log_{2}{x} \), produce all real numbers from negative infinity to positive infinity as \( x \) moves over its domain. This means:

• For very small values of \( x \) just greater than zero (e.g., close to zero, like 0.0001), \( \log_{2}{x} \) will result in large negative numbers.
• Conversely, for large \( x \), the logarithm produces large positive numbers.

Adding \( \frac{1}{3} \) to \( \log_{2}{x} \) simply shifts the entire function's output values upward by \( \frac{1}{3} \) units. However, this shift does not alter the fundamental range, which still stretches to cover all real numbers. So, the range remains all real numbers, ensuring the function can potentially take any real number as a result.

Transformation of Functions

Transformations are ways to modify the appearance and behavior of a function on a graph without changing its fundamental characteristics. Understanding transformations can help you quickly determine how a function like \( y = \log_{2}{x} + \frac{1}{3} \) will behave or look compared to its basic form \( y = \log_{2}{x} \).

There are different types of transformations that can be applied:

• Vertical Shift: Adding \( \frac{1}{3} \) results in a vertical upward shift of the whole graph by \( \frac{1}{3} \) units. This happens because every output value, \( y \), is consistently increased by \( \frac{1}{3} \).
• Horizontal Shifts, Stretches, and Reflects: Although not present in this specific function, these transformations involve moving the graph left or right, stretching or compressing it, or flipping it across an axis.

Understanding these transformations helps predict and visualize the effect of different mathematical operations on functions. The vertical shift does not alter the domain or range of \( y = \log_{2}{x} + \frac{1}{3} \), but it does change the specific y-values associated with each x-value. Hence, transformations are a powerful tool in shaping and modifying functions to get the desired output or visualize how the function changes relative to its basic form.