Answer :
To solve the problem related to the given function [tex]\( f(x) = \log_3 (x + 3) \)[/tex], let's analyze and understand it step by step.
### Step-by-Step Solution
1. Understanding the Function
- The function [tex]\( f(x) = \log_3 (x + 3) \)[/tex] represents the logarithm of [tex]\( x + 3 \)[/tex] with base 3. Logarithms tell us the power to which a base number must be raised to obtain another number.
2. Domain of the Function
- The argument inside the logarithm, [tex]\( x + 3 \)[/tex], must be greater than 0 because logarithms are only defined for positive numbers.
- Therefore, [tex]\( x + 3 > 0 \)[/tex] which simplifies to [tex]\( x > -3 \)[/tex].
- Domain: [tex]\( x > -3 \)[/tex]
3. Evaluation Examples
- To better understand the function, let's evaluate it for some specific values of [tex]\( x \)[/tex]:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \log_3 (0 + 3) = \log_3 3 = 1 \quad \text{(since } 3^1 = 3\text{)} \][/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[ f(6) = \log_3 (6 + 3) = \log_3 9 = 2 \quad \text{(since } 3^2 = 9\text{)} \][/tex]
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = \log_3 (-2 + 3) = \log_3 1 = 0 \quad \text{(since } 3^0 = 1\text{)} \][/tex]
4. Graph of the Function
- The function can be graphed with [tex]\( x \)[/tex] on the horizontal axis and [tex]\( f(x) \)[/tex] on the vertical axis.
- As [tex]\( x \)[/tex] approaches [tex]\(-3\)[/tex] from the right (i.e., [tex]\( x \to -3^+ \)[/tex]), [tex]\( x + 3 \to 0^+ \)[/tex] and consequently [tex]\( f(x) \to -\infty \)[/tex].
- As [tex]\( x \to \infty \)[/tex], [tex]\( x + 3 \to \infty \)[/tex] and therefore [tex]\( f(x) \to \infty \)[/tex] slowly.
- The graph will continuously increase but at a decreasing rate as [tex]\( x \)[/tex] increases.
5. Derivative of the Function [tex]\( f'(x) \)[/tex]
- The derivative of [tex]\( f(x) \)[/tex] can be found using the chain rule and the property of logarithms. For [tex]\( f(x) = \log_3 (x + 3) \)[/tex]:
[tex]\[ f'(x) = \frac{1}{\ln(3)} \cdot \frac{d}{dx} \ln(x + 3) = \frac{1}{\ln(3)} \cdot \frac{1}{x+3} \][/tex]
Summary:
The function [tex]\( f(x) = \log_3 (x + 3) \)[/tex] is defined for [tex]\( x > -3 \)[/tex] and tells us the power to which 3 must be raised to get [tex]\( x + 3 \)[/tex]. Key points and properties have been discussed with examples and graphical behavior, ensuring a comprehensive understanding of the function.
### Step-by-Step Solution
1. Understanding the Function
- The function [tex]\( f(x) = \log_3 (x + 3) \)[/tex] represents the logarithm of [tex]\( x + 3 \)[/tex] with base 3. Logarithms tell us the power to which a base number must be raised to obtain another number.
2. Domain of the Function
- The argument inside the logarithm, [tex]\( x + 3 \)[/tex], must be greater than 0 because logarithms are only defined for positive numbers.
- Therefore, [tex]\( x + 3 > 0 \)[/tex] which simplifies to [tex]\( x > -3 \)[/tex].
- Domain: [tex]\( x > -3 \)[/tex]
3. Evaluation Examples
- To better understand the function, let's evaluate it for some specific values of [tex]\( x \)[/tex]:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \log_3 (0 + 3) = \log_3 3 = 1 \quad \text{(since } 3^1 = 3\text{)} \][/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[ f(6) = \log_3 (6 + 3) = \log_3 9 = 2 \quad \text{(since } 3^2 = 9\text{)} \][/tex]
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = \log_3 (-2 + 3) = \log_3 1 = 0 \quad \text{(since } 3^0 = 1\text{)} \][/tex]
4. Graph of the Function
- The function can be graphed with [tex]\( x \)[/tex] on the horizontal axis and [tex]\( f(x) \)[/tex] on the vertical axis.
- As [tex]\( x \)[/tex] approaches [tex]\(-3\)[/tex] from the right (i.e., [tex]\( x \to -3^+ \)[/tex]), [tex]\( x + 3 \to 0^+ \)[/tex] and consequently [tex]\( f(x) \to -\infty \)[/tex].
- As [tex]\( x \to \infty \)[/tex], [tex]\( x + 3 \to \infty \)[/tex] and therefore [tex]\( f(x) \to \infty \)[/tex] slowly.
- The graph will continuously increase but at a decreasing rate as [tex]\( x \)[/tex] increases.
5. Derivative of the Function [tex]\( f'(x) \)[/tex]
- The derivative of [tex]\( f(x) \)[/tex] can be found using the chain rule and the property of logarithms. For [tex]\( f(x) = \log_3 (x + 3) \)[/tex]:
[tex]\[ f'(x) = \frac{1}{\ln(3)} \cdot \frac{d}{dx} \ln(x + 3) = \frac{1}{\ln(3)} \cdot \frac{1}{x+3} \][/tex]
Summary:
The function [tex]\( f(x) = \log_3 (x + 3) \)[/tex] is defined for [tex]\( x > -3 \)[/tex] and tells us the power to which 3 must be raised to get [tex]\( x + 3 \)[/tex]. Key points and properties have been discussed with examples and graphical behavior, ensuring a comprehensive understanding of the function.
