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To determine which is the graph of the function [tex]\( f(x) = \log(x + 1) - 4 \)[/tex], let us analyze the function step-by-step.
### Understand the Function
1. Domain Analysis:
- The function involves [tex]\( \log(x + 1) \)[/tex]. The logarithm is defined for positive arguments, thus [tex]\( x + 1 > 0 \Rightarrow x > -1 \)[/tex].
- Therefore, the domain of [tex]\( f(x) \)[/tex] is [tex]\( (-1, \infty) \)[/tex].
2. Initial Value Calculations:
- Let's start by evaluating [tex]\( f(x) \)[/tex] at a few critical points to understand its behavior:
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \log(0 + 1) - 4 = \log(1) - 4 = 0 - 4 = -4 \][/tex]
- As [tex]\( x \)[/tex] approaches [tex]\( -1 \)[/tex] from the right (i.e., [tex]\( x \to -1^+ \)[/tex]):
[tex]\[ f(x) \to \log(0) - 4 \][/tex]
- Since [tex]\( \log(0) \)[/tex] approaches [tex]\(-\infty\)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\(-\infty\)[/tex].
3. Behavior at Infinity:
- As [tex]\( x \)[/tex] becomes very large (i.e., [tex]\( x \to \infty \)[/tex]):
[tex]\[ f(x) \to \log(\infty) - 4 \][/tex]
- Since [tex]\( \log(\infty) \)[/tex] approaches [tex]\( \infty \)[/tex], [tex]\( f(x) \)[/tex] also approaches [tex]\(\infty\)[/tex].
4. Graph Characteristics:
- The function [tex]\( f(x) = \log(x + 1) - 4 \)[/tex] is a logarithmic function shifted downwards by 4 units.
- It has a vertical asymptote at [tex]\( x = -1 \)[/tex], where it drops off to [tex]\(-\infty\)[/tex].
- It crosses the y-axis at [tex]\( (0, -4) \)[/tex].
- As [tex]\( x \)[/tex] increases, the function rises without bound.
### Graph Analysis
To match the function with its graph:
1. Vertical Asymptote: The graph should have a vertical asymptote at [tex]\( x = -1 \)[/tex].
2. Y-intercept: The graph should intersect the y-axis at [tex]\( (0, -4) \)[/tex].
3. Increasing Behavior: After crossing the y-axis, the graph should increase slowly without bound.
By considering these characteristics, we can identify the correct graph. The graph of [tex]\( f(x) = \log(x + 1) - 4 \)[/tex] should start from very low values near [tex]\( x = -1 \)[/tex], cross the y-axis at [tex]\( -4 \)[/tex], and increase gradually as [tex]\( x \)[/tex] becomes large.
This detailed analysis should help you identify the correct graph from the given options.
### Understand the Function
1. Domain Analysis:
- The function involves [tex]\( \log(x + 1) \)[/tex]. The logarithm is defined for positive arguments, thus [tex]\( x + 1 > 0 \Rightarrow x > -1 \)[/tex].
- Therefore, the domain of [tex]\( f(x) \)[/tex] is [tex]\( (-1, \infty) \)[/tex].
2. Initial Value Calculations:
- Let's start by evaluating [tex]\( f(x) \)[/tex] at a few critical points to understand its behavior:
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \log(0 + 1) - 4 = \log(1) - 4 = 0 - 4 = -4 \][/tex]
- As [tex]\( x \)[/tex] approaches [tex]\( -1 \)[/tex] from the right (i.e., [tex]\( x \to -1^+ \)[/tex]):
[tex]\[ f(x) \to \log(0) - 4 \][/tex]
- Since [tex]\( \log(0) \)[/tex] approaches [tex]\(-\infty\)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\(-\infty\)[/tex].
3. Behavior at Infinity:
- As [tex]\( x \)[/tex] becomes very large (i.e., [tex]\( x \to \infty \)[/tex]):
[tex]\[ f(x) \to \log(\infty) - 4 \][/tex]
- Since [tex]\( \log(\infty) \)[/tex] approaches [tex]\( \infty \)[/tex], [tex]\( f(x) \)[/tex] also approaches [tex]\(\infty\)[/tex].
4. Graph Characteristics:
- The function [tex]\( f(x) = \log(x + 1) - 4 \)[/tex] is a logarithmic function shifted downwards by 4 units.
- It has a vertical asymptote at [tex]\( x = -1 \)[/tex], where it drops off to [tex]\(-\infty\)[/tex].
- It crosses the y-axis at [tex]\( (0, -4) \)[/tex].
- As [tex]\( x \)[/tex] increases, the function rises without bound.
### Graph Analysis
To match the function with its graph:
1. Vertical Asymptote: The graph should have a vertical asymptote at [tex]\( x = -1 \)[/tex].
2. Y-intercept: The graph should intersect the y-axis at [tex]\( (0, -4) \)[/tex].
3. Increasing Behavior: After crossing the y-axis, the graph should increase slowly without bound.
By considering these characteristics, we can identify the correct graph. The graph of [tex]\( f(x) = \log(x + 1) - 4 \)[/tex] should start from very low values near [tex]\( x = -1 \)[/tex], cross the y-axis at [tex]\( -4 \)[/tex], and increase gradually as [tex]\( x \)[/tex] becomes large.
This detailed analysis should help you identify the correct graph from the given options.
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