Domain, Range, and Graph of f(x) = log(x - 1)

Math Problem Statement

Use the given function f to answer parts (a) through (c) below.

f left parenthesis x right parenthesis equals log left parenthesis x minus 1 right parenthesis minus 5f(x)=log(x−1)−5

Question content area bottom

Part 1

(a) Find the domain of f.

The domain of f is

enter your response here.

(Type your answer in interval notation.)

(b) Graph f. Choose the correct graph below.

-1010-100xy

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 2 and a vertical y-axis labeled from negative 10 to 0 in increments of 1. From left to right, a curve rises steeply until it reaches (2, negative 5), then rises shallowly passing through the points (4, negative 4.5) (5, negative 4.4). As x approaches 1, the curve approaches negative infinity. All coordinates are approximate.

-1010010xy

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 2 and a vertical y-axis labeled from 0 to 10 in increments of 1. From left to right, a curve rises steeply until it reaches (2, 5), then rises shallowly passing through the points (4, 5.5) and (5, 5.6). As x approaches 1, the curve approaches negative infinity. All coordinates are approximate.

-1010-100xy

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 2 and a vertical y-axis labeled from negative 10 to 0 in increments of 1. From left to right, a curve rises steeply until it reaches (0, negative 5), then rises shallowly passing through the points (2, negative 4.5) and (3, negative 4.4). As x approaches negative 1, the curve approaches negative infinity. All coordinates are approximate.

-1010010xy

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 2 and a vertical y-axis labeled from 0 to 10 in increments of 1. From left to right, a curve rises steeply until it reaches (0, 5), then rises shallowly passing through the points (2, 5.5) and (3, 5.6). As x approaches negative 1, the curve approaches negative infinity. All coordinates are approximate.

The range of f is

enter your response here.

(Type your answer in interval notation.)

Solution

Given the function $f(x) = \log(x - 1) - 5$ , let's work through each part step-by-step.

Part (a): Domain of $f$

The function $f(x)$ contains a logarithmic expression, $\log(x - 1)$ . For the logarithmic function to be defined, the argument inside the logarithm must be positive:

$x - 1 > 0$

Solving for $x$ , we find:

$x > 1$

Therefore, the domain of $f$ is $(1, \infty)$ .

Answer for (a): The domain of $f$ is $(1, \infty)$ .

Part (b): Graph of $f$

The function $f(x) = \log(x - 1) - 5$ is a transformation of the basic logarithmic function $\log(x)$ :

The $x - 1$ inside the logarithm shifts the graph to the right by 1 unit.
The $-5$ outside the logarithm shifts the graph down by 5 units.

From this, we expect the graph to:

Have a vertical asymptote at $x = 1$ , since $\log(x - 1) \to -\infty$ as $x \to 1^+$ .
Pass through points such as $(2, -5)$ , since $f(2) = \log(2 - 1) - 5 = -5$ .

Based on the descriptions provided:

Option A shows the correct behavior, with a vertical asymptote at $x = 1$ and the function values decreasing to $-\infty$ as $x$ approaches 1 from the right.

Answer for (b): The correct graph is A.

Part (c): Range of $f$

From the graph (Option A), we observe that:

As $x \to 1^+$ , $f(x) \to -\infty$ .
As $x \to \infty$ , $f(x) \to \infty$ , since $\log(x - 1)$ continues to increase without bound.

Thus, the range of $f$ is all real numbers.

Answer for (c): The range of $f$ is $(-\infty, \infty)$ .

Would you like further clarification on any of these parts?

Relative questions to expand understanding:

What transformations affect the graph of a logarithmic function and how?
How would the domain change if the function were $\log(x + 1) - 5$ ?
Why does the function have a vertical asymptote at $x = 1$ ?
How would the range change if the function were $-\log(x - 1) - 5$ ?
How does the base of the logarithm affect the shape of the graph?

Tip: For functions with logarithms, always check the domain carefully to avoid undefined values.

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Math Problem Analysis

Mathematical Concepts

Logarithmic Functions
Domain and Range
Graph Transformations

Formulas

f(x) = log(x - 1) - 5

Theorems

Domain of logarithmic functions
Range of logarithmic functions
Transformation of graphs of logarithmic functions

Suitable Grade Level

Grades 10-12

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