Chapter 17: Problem 57
For what value of \(g(x)\) is the function of \(f(x)=g(x)\) ? (a) \(g(x)=e\) (b) \(g(x)=e^{x}\) (c) \(g(x)=\log x\) (d) none of these
Short Answer
Expert verified
Answer: (a) \(g(x)=e\)
Step by step solution
01
Check answer choice (a) \(g(x)=e\)
To check if the given expression for \(g(x)\), \(g(x)=e\), satisfies the condition \(f(x)=g(x)\), we need to determine if the following equation is true: \(f(x)=g(x)=e\). Considering there is no variable in the equation, it is a constant function \(f(x)=e\) for all \(x\) in its domain which equals to \(g(x)\). Hence, \(f(x)=g(x)\) for \(g(x)=e\).
02
Check answer choice (b) \(g(x)=e^{x}\)
Now, let's examine the second answer choice, where \(g(x)=e^{x}\). To determine if this expression satisfies \(f(x) = g(x)\), we need to compare \(f(x)\) with \(g(x)\). In this case, we cannot claim that \(f(x)\) is always equal to \(g(x)\) because \(f(x)\) is not clearly stated. We only know \(f(x)=g(x)\), and there is no other information about \(f(x)\) that can lead to \(f(x)=e^{x}\). Therefore, we cannot claim that \(f(x)=g(x)\) for \(g(x)=e^{x}\).
03
Check answer choice (c) \(g(x)=\log x\)
Let's examine the third answer choice, where \(g(x)=\log x\). To determine if this expression satisfies \(f(x) = g(x)\), we need to compare \(f(x)\) with \(g(x)\). Similar to the previous step, since no information about \(f(x)\) is given, we cannot claim that \(f(x)=g(x)\) for \(g(x)=\log x\).
04
Check answer choice (d) none of these
As we have gone through the possible answer choices (a), (b), and (c), we found that the only expression which satisfied the condition \(f(x)=g(x)\) was \(g(x)=e\). Therefore, the correct answer is not "none of these", since there is a valid expression for \(g(x)\) that makes \(f(x)=g(x)\).
05
Conclusion
Based on the analysis of each answer choice, we conclude that the value of \(g(x)\) that makes the function of \(f(x)=g(x)\) is \(g(x)=e\). So the correct answer is choice (a).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Constant Function
A constant function is a function that always returns the same value, no matter what the input is. This type of function is represented as \( f(x) = c \), where \( c \) is a constant value. For example, if \( f(x) = e \) for all values of \( x \), then \( f(x) \) is a constant function. That means whether \( x \) is 1, 2, or 100, the output will always be \( e \).
- A constant function is simple and easy to understand.
- The graph of a constant function is a horizontal line.
- The value of \( f(x) \) does not change as \( x \) changes.
Exponential Function
An exponential function is a function where a constant base is raised to the power of the variable. These are represented as \( f(x) = a^x \), where \( a \) is a constant. The most common exponential function is \( f(x) = e^x \). Exponential functions exhibit rapid growth or decay depending on the value of the base.
- The graph of an exponential function grows wondersomely fast.
- It forms a curve that continuously increases or decreases.
- In situations involving growth, such as population growth, or in decay, such as radioactive decay, exponential functions are often used.
Logarithmic Function
A logarithmic function is the inverse of an exponential function, expressed as \( f(x) = \log_b(x) \), where \( b \) is the base of the logarithm. Commonly, the base 10 logarithm \( \log(x) \) and the natural logarithm \( \ln(x) \) are used, especially in mathematical calculations. Logarithmic functions grow slowly and are useful in many fields such as science, engineering, and statistics.
- The graph of a logarithmic function increases steadily but never touches the axes.
- They are particularly helpful in situations where we need to compress data or deal with large ranges of values.
- Logarithmic functions simplify multiplication into addition, which is useful for computational purposes.