Consider the following in respect of the function f(x) = 10 x: 1. Its domain is (-∞, ∞) 2. It is a continuous function 3. It is differentiable at x = 0 Which of the above statements are correct?
1, 2 and 3
Analyzing Properties of the Function f(x) = 10x
Let's examine the properties of the function given, \(f(x) = 10^x\), based on the statements provided.
Understanding the Domain of f(x) = 10x
The domain of a function is the set of all possible input values (x) for which the function is defined. For an exponential function of the form \(a^x\), where the base \(a\) is a positive real number other than 1, the function is defined for all real numbers. In this case, the base is 10, which is positive and not equal to 1.
Therefore, the function \(f(x) = 10^x\) is defined for every real number \(x\).
- Statement 1: Its domain is \((-\infty, \infty)\)
- This statement says the domain is all real numbers, which is true for \(f(x) = 10^x\).
- Thus, statement 1 is correct.
Examining the Continuity of f(x) = 10x
A continuous function is one where small changes in the input result in small changes in the output. Graphically, a continuous function can be drawn without lifting the pen. Exponential functions of the form \(a^x\), with \(a > 0\) and \(a \neq 1\), are known to be continuous over their entire domain.
Since the domain of \(f(x) = 10^x\) is \((-\infty, \infty)\), and it is an exponential function with a positive base not equal to 1, it is continuous everywhere on its domain.
- Statement 2: It is a continuous function
- This statement claims the function is continuous. Since it's continuous over its entire domain \((-\infty, \infty)\), this statement is correct.
- Thus, statement 2 is correct.
Investigating the Differentiability of f(x) = 10x at x = 0
A function is differentiable at a point if its derivative exists at that point. The derivative of a function represents the instantaneous rate of change. For an exponential function of the form \(f(x) = a^x\), where \(a > 0\) and \(a \neq 1\), the derivative is given by \(f'(x) = a^x \ln(a)\).
For the function \(f(x) = 10^x\), the derivative is \(f'(x) = 10^x \ln(10)\).
This derivative formula \(f'(x) = 10^x \ln(10)\) is defined for all real numbers \(x\), because \(10^x\) is defined for all real \(x\) and \(\ln(10)\) is a constant.
To check differentiability at \(x=0\), we evaluate the derivative at \(x=0\):
\(f'(0) = 10^0 \ln(10)\)
\(f'(0) = 1 \cdot \ln(10)\)
\(f'(0) = \ln(10)\)
Since \(\ln(10)\) is a well-defined real number (approximately 2.3026), the derivative exists at \(x=0\).
- Statement 3: It is differentiable at x = 0
- This statement claims the function is differentiable at \(x=0\). Since the derivative \(f'(0) = \ln(10)\) exists, this statement is correct.
- Thus, statement 3 is correct.
Summary of Findings
We have analyzed each statement regarding the function \(f(x) = 10^x\):
| Statement | Description | Correctness |
|---|---|---|
| 1 | Domain is \((-\infty, \infty)\) | Correct |
| 2 | Is a continuous function | Correct |
| 3 | Is differentiable at \(x=0\) | Correct |
Based on our analysis, all three statements are correct.
Revision Table: Function Properties
| Property | f(x) = 10x | Explanation |
|---|---|---|
| Domain | \((-\infty, \infty)\) | Defined for all real numbers x. |
| Range | \((0, \infty)\) | Output values are always positive. |
| Continuity | Continuous everywhere | Exponential functions \(a^x\) (\(a>0, a \neq 1\)) are continuous on their domain. |
| Differentiability | Differentiable everywhere | Derivative \(f'(x) = 10^x \ln(10)\) exists for all real x. |
| Derivative at x=0 | \(\ln(10)\) | \(f'(0) = 10^0 \ln(10) = \ln(10)\). |
Additional Information: Exponential Functions
Exponential functions of the form \(f(x) = a^x\), where \(a\) is a positive constant and \(a \neq 1\), have several key properties:
- The base \(a\) must be positive. If \(a\) were negative, the function might not be defined for all real numbers (e.g., \((-4)^{1/2}\) is not a real number). If \(a=1\), \(f(x)=1^x=1\), which is just a constant function.
- The domain is always \((-\infty, \infty)\).
- The range is always \((0, \infty)\). The graph is always above the x-axis.
- The graph always passes through the point \((0, 1)\) because \(a^0 = 1\) for any \(a \neq 0\).
- If \(a > 1\), the function is strictly increasing. The graph rises from left to right.
- If \(0 < a < 1\), the function is strictly decreasing. The graph falls from left to right.
- Exponential functions with base \(a > 0, a \neq 1\) are both continuous and differentiable for all real numbers.
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