Problem 64 - Precalculus

Question

Sketch the graph of \(y=g(x)\) by starting with the graph of \(y=f(x)\) and using transformations. Track at least three points of your choice and the vertical asymptote through the transformations. State the domain and range of \(g\). \(f(x)=\log _{2}(x), g(x)=\log _{2}(x+1)\)

Step-by-Step Solution

Answer

Domain of \(g\) is \((-1, \infty)\) and the range is \((-\infty, \infty)\). Vertical asymptote shifts to \(x = -1\).

Step 1: Understand the Original Function

The original function, \(f(x) = \log_{2}(x)\), is a logarithmic function with base 2. It has a vertical asymptote at \(x=0\), and its domain is \((0, \infty)\). The range is all real numbers, \((-\infty, \infty)\).

Step 2: Identify the Transformation

The transformation in \(g(x) = \log_{2}(x+1)\) is a horizontal shift. Specifically, adding 1 to \(x\) in the argument of the logarithm function shifts the graph of \(f(x)\) one unit to the left.

Step 3: Track a Few Points

Choose three points and the vertical asymptote on \(f(x)\):1. Point \((1, 0)\) because \(\log_{2}(1) = 0\).2. Point \((2, 1)\) because \(\log_{2}(2) = 1\).3. Point \((4, 2)\) because \(\log_{2}(4) = 2\).4. Vertical Asymptote at \(x=0\).Apply the transformation:1. \((1, 0)\) shifts to \((0, 0)\).2. \((2, 1)\) shifts to \((1, 1)\).3. \((4, 2)\) shifts to \((3, 2)\).4. The vertical asymptote at \(x=0\) shifts to \(x = -1\).

Step 4: Determine the Domain and Range of \(g(x)\)

The domain of \(g(x) = \log_{2}(x+1)\) changes because of the shift. It is \((-1, \infty)\) as you need \(x + 1 > 0\), hence \(x > -1\). The range remains \((-, \infty)\) because logarithms can still take any real value.

Key Concepts

Understanding Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. If you have an exponential function like \( y = 2^x \), the corresponding logarithmic function would be \( x = \log_2(y) \).
Logarithmic functions help you determine the power to which a given base (in this case, 2) must be raised to produce a certain number.

• The base of a logarithm is always positive and greater than 1.
• The function \( f(x) = \log_{2}(x) \) implies we're working with a logarithm of base 2.
• A property of logarithmic functions is that they have a vertical asymptote, where the function tends towards negative infinity.

These functions typically start undefined at zero, because you cannot raise a base to any power and get zero as a result. Hence, the domain does not include zero or any negative numbers. Understanding this helps when predicting graph behaviors and transformations.

The Domain and Range of Logarithmic Functions

The domain of a function defines all the input values (or x-values) the function can accept, while the range defines all possible outputs (or y-values).
For the function \( f(x) = \log_{2}(x) \), the domain is \((0, \infty)\). This means the function only accepts positive values of \( x \).

• Logarithms are not defined for zero or negative numbers, hence the domain starts just above zero.
• The range for logarithmic functions, such as this one, is \((-\infty, \infty)\) because the output can be any real number.

When we consider \( g(x) = \log_{2}(x+1) \), the graph shifts horizontally to the left by one unit, thus altering the domain to \((-1, \infty)\) while keeping the range unchanged.

Exploring Vertical Asymptotes in Logarithmic Graphs

Vertical asymptotes in graphs of logarithmic functions are lines that the graph approaches but never actually touches or crosses.
In the function \( f(x) = \log_{2}(x) \), the vertical asymptote is at \( x = 0 \). This is because as \( x \) approaches zero from the right, the logarithm approaches negative infinity.

• Vertical asymptotes signify where the function is undefined at certain points.
• They are crucial for understanding the limits and behavior of a graph as inputs reach these critical values.

With the transformation to \( g(x) = \log_{2}(x+1) \), this asymptote shifts to \( x = -1 \). Tracking these shifts is integral for accurately sketching the transformed graph and understanding the function's behavior.