Graph each function. ƒ(x) = log3 (x-1) + 2

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

Problem 55

Chapter 5, Problem 55

Graph each function. ƒ(x) = log₃ (x-1) + 2

Verified step by step guidance

Identify the base of the logarithm and the transformations applied to the function. Here, the function is $f(x) = \log_{3}(x - 1) + 2$, which is a logarithmic function with base 3, shifted horizontally and vertically.

Determine the domain of the function by setting the argument of the logarithm greater than zero: $x - 1 > 0$. Solve this inequality to find the domain.

Find the vertical asymptote by setting the argument of the logarithm equal to zero: $x - 1 = 0$. This vertical line is where the function approaches infinity or negative infinity.

Identify the horizontal shift and vertical shift. The $x - 1$ inside the logarithm shifts the graph 1 unit to the right, and the $+2$ outside shifts the graph 2 units upward.

Plot key points by choosing values of $x$ greater than 1, calculate corresponding $f(x)$ values using the function, and sketch the graph showing the vertical asymptote at $x=1$, the general shape of the logarithmic curve, and the shifts.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logarithmic Functions

A logarithmic function is the inverse of an exponential function and is written as f(x) = log_b(x), where b is the base. It answers the question: to what power must the base b be raised to produce x? Understanding the properties of logarithms is essential for graphing and interpreting these functions.

Domain of Logarithmic Functions

The domain of a logarithmic function f(x) = log_b(x - h) is all x-values for which the argument (x - h) is positive. For example, in f(x) = log_3(x - 1) + 2, the domain is x > 1 because the logarithm is undefined for zero or negative inputs. Identifying the domain helps in correctly plotting the graph.

Transformations of Functions

Transformations shift or change the shape of a graph. For f(x) = log_3(x - 1) + 2, the (x - 1) inside the log shifts the graph 1 unit to the right, and the +2 outside shifts it 2 units up. Recognizing horizontal and vertical shifts is crucial for accurately graphing the function.

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