Answer :
To determine the graph of the function [tex]\( f(x) = \log(x+1) - 4 \)[/tex], we need to analyze its general shape and important properties.
### Step-by-Step Analysis:
1. Function Definition:
[tex]\[ f(x) = \log(x+1) - 4 \][/tex]
Here, [tex]\(\log\)[/tex] refers to the natural logarithm (base [tex]\(e\)[/tex]).
2. Domain:
The logarithmic function [tex]\(\log(x+1)\)[/tex] is defined when [tex]\(x + 1 > 0\)[/tex], i.e., [tex]\(x > -1\)[/tex]. Therefore, the domain of [tex]\(f(x)\)[/tex] is [tex]\((-1, \infty)\)[/tex].
3. Vertical Asymptote:
The function [tex]\(\log(x+1)\)[/tex] approaches [tex]\(-\infty\)[/tex] as [tex]\(x \)[/tex] approaches [tex]\(-1\)[/tex] from the right. Thus, [tex]\(f(x)\)[/tex] has a vertical asymptote at [tex]\(x = -1\)[/tex].
4. Shifts and Transformations:
- Horizontal Shift: The [tex]\(\log(x+1)\)[/tex] part indicates a shift 1 unit to the left compared to the basic [tex]\(\log x\)[/tex] function.
- Vertical Shift: Subtracting 4 from [tex]\(\log(x+1)\)[/tex] shifts the graph 4 units downward.
5. Behavior as [tex]\(x \to \infty\)[/tex]:
As [tex]\(x\)[/tex] increases, [tex]\(\log(x+1)\)[/tex] increases without bound, making [tex]\(\log(x+1) - 4\)[/tex] also increase without bound (towards [tex]\(\infty\)[/tex]).
6. Intercepts:
- x-intercept: To find the x-intercept, solve [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ \log(x+1) - 4 = 0 \implies \log(x+1) = 4 \implies x + 1 = e^4 \implies x = e^4 - 1 \][/tex]
- y-intercept: To find the y-intercept, evaluate [tex]\(f(0)\)[/tex]:
[tex]\[ f(0) = \log(0 + 1) - 4 = \log(1) - 4 = 0 - 4 = -4 \][/tex]
So, the y-intercept is [tex]\((0, -4)\)[/tex].
7. Summary of Key Points:
- Domain: [tex]\(x > -1\)[/tex]
- Vertical asymptote: [tex]\(x = -1\)[/tex]
- x-intercept at [tex]\(x = e^4 - 1\)[/tex]
- y-intercept [tex]\((0, -4)\)[/tex]
- Increases without bound as [tex]\(x \to \infty\)[/tex]
### Sketching the Graph:
- Draw a vertical asymptote at [tex]\(x = -1\)[/tex].
- Mark the y-intercept [tex]\((0, -4)\)[/tex] and the x-intercept [tex]\( \left(e^4 - 1, 0\right) \)[/tex].
- As [tex]\(x \rightarrow -1\)[/tex] from the right, [tex]\(f(x) \rightarrow -\infty\)[/tex].
- As [tex]\(x \rightarrow \infty\)[/tex], [tex]\(f(x) \rightarrow \infty\)[/tex].
- The function is shifted 1 unit left and 4 units down from the basic logarithmic function.
By carefully plotting these details, you can visualize the graph of [tex]\( f(x) = \log(x+1) - 4 \)[/tex] correctly.
### Step-by-Step Analysis:
1. Function Definition:
[tex]\[ f(x) = \log(x+1) - 4 \][/tex]
Here, [tex]\(\log\)[/tex] refers to the natural logarithm (base [tex]\(e\)[/tex]).
2. Domain:
The logarithmic function [tex]\(\log(x+1)\)[/tex] is defined when [tex]\(x + 1 > 0\)[/tex], i.e., [tex]\(x > -1\)[/tex]. Therefore, the domain of [tex]\(f(x)\)[/tex] is [tex]\((-1, \infty)\)[/tex].
3. Vertical Asymptote:
The function [tex]\(\log(x+1)\)[/tex] approaches [tex]\(-\infty\)[/tex] as [tex]\(x \)[/tex] approaches [tex]\(-1\)[/tex] from the right. Thus, [tex]\(f(x)\)[/tex] has a vertical asymptote at [tex]\(x = -1\)[/tex].
4. Shifts and Transformations:
- Horizontal Shift: The [tex]\(\log(x+1)\)[/tex] part indicates a shift 1 unit to the left compared to the basic [tex]\(\log x\)[/tex] function.
- Vertical Shift: Subtracting 4 from [tex]\(\log(x+1)\)[/tex] shifts the graph 4 units downward.
5. Behavior as [tex]\(x \to \infty\)[/tex]:
As [tex]\(x\)[/tex] increases, [tex]\(\log(x+1)\)[/tex] increases without bound, making [tex]\(\log(x+1) - 4\)[/tex] also increase without bound (towards [tex]\(\infty\)[/tex]).
6. Intercepts:
- x-intercept: To find the x-intercept, solve [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ \log(x+1) - 4 = 0 \implies \log(x+1) = 4 \implies x + 1 = e^4 \implies x = e^4 - 1 \][/tex]
- y-intercept: To find the y-intercept, evaluate [tex]\(f(0)\)[/tex]:
[tex]\[ f(0) = \log(0 + 1) - 4 = \log(1) - 4 = 0 - 4 = -4 \][/tex]
So, the y-intercept is [tex]\((0, -4)\)[/tex].
7. Summary of Key Points:
- Domain: [tex]\(x > -1\)[/tex]
- Vertical asymptote: [tex]\(x = -1\)[/tex]
- x-intercept at [tex]\(x = e^4 - 1\)[/tex]
- y-intercept [tex]\((0, -4)\)[/tex]
- Increases without bound as [tex]\(x \to \infty\)[/tex]
### Sketching the Graph:
- Draw a vertical asymptote at [tex]\(x = -1\)[/tex].
- Mark the y-intercept [tex]\((0, -4)\)[/tex] and the x-intercept [tex]\( \left(e^4 - 1, 0\right) \)[/tex].
- As [tex]\(x \rightarrow -1\)[/tex] from the right, [tex]\(f(x) \rightarrow -\infty\)[/tex].
- As [tex]\(x \rightarrow \infty\)[/tex], [tex]\(f(x) \rightarrow \infty\)[/tex].
- The function is shifted 1 unit left and 4 units down from the basic logarithmic function.
By carefully plotting these details, you can visualize the graph of [tex]\( f(x) = \log(x+1) - 4 \)[/tex] correctly.
