Graph each function. ƒ(x) = log￬6 (x-2)

Verified step by step guidance

1. Identify the base of the logarithm. In this case, the base is 6, which means the function is a logarithm with base 6.
2. Recognize the transformation in the function. The expression inside the logarithm is (x-2), indicating a horizontal shift to the right by 2 units.
3. Determine the domain of the function. Since the logarithm is undefined for non-positive numbers, set the inside of the logarithm greater than zero: x - 2 > 0, which simplifies to x > 2.
4. Identify the vertical asymptote. The vertical asymptote occurs where the argument of the logarithm is zero, which is at x = 2.
5. Sketch the graph. Start by plotting the vertical asymptote at x = 2, then plot points for values of x greater than 2, and draw the curve approaching the asymptote from the right, increasing slowly as x increases.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. They are defined for positive real numbers and have a base, which in this case is 6. The function ƒ(x) = log₆(x-2) indicates that we are looking for the power to which the base 6 must be raised to obtain the value of (x-2). Understanding the properties of logarithms, such as their domain and range, is essential for graphing them accurately.

Domain and Range

The domain of a function refers to all possible input values (x-values) that the function can accept, while the range refers to all possible output values (y-values). For the function ƒ(x) = log₆(x-2), the domain is x > 2, since the argument of the logarithm must be positive. The range of logarithmic functions is all real numbers, which means the graph will extend infinitely in the vertical direction.

Graphing Techniques

Graphing techniques involve plotting points and understanding the shape of the function based on its properties. For logarithmic functions, the graph typically passes through the point (2, 0) and approaches the vertical asymptote at x = 2. Familiarity with transformations, such as shifts and reflections, is crucial for accurately representing the function on a coordinate plane.