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Chapter 6: Problem 11

True or False The function \(f(x)=e^{x}\) is increasing and is one-to-one.

Short Answer

Expert verified
True

Step by step solution

01

Understand the Function

Consider the function given: \[ f(x) = e^x \]where \( e \) is Euler's number, approximately equal to 2.718.
02

Check If the Function is Increasing

An increasing function has the property that if \( x_1 < x_2 \), then \( f(x_1) < f(x_2) \). For the function \( f(x) = e^x \), note that the derivative \( f'(x) = e^x \) is always positive for all real numbers \( x \). Thus, \( f(x) \) is always increasing.
03

Determine If the Function is One-to-One

A function is one-to-one if every element in the range corresponds to exactly one element in the domain. Since we have established that \( f(x) = e^x \) is strictly increasing, it means that it never takes the same value twice. Therefore, it is one-to-one.
04

Conclusion

Since we have shown that \( f(x) = e^x \) is both increasing and one-to-one, the statement is true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Increasing Functions
An increasing function has the property that as the input (or x-value) increases, the output (or y-value) also increases. Mathematically, if you have two numbers such that \(x_1 < x_2\), then for an increasing function \(f(x)\), \(f(x_1) < f(x_2)\). In simple terms, as you move from left to right on a graph of an increasing function, the line always goes upwards.
For the exponential function \(f(x) = e^x\), its derivative is \(f'(x) = e^x\). Since \(e^x\) is always positive, the function is always increasing. This means that no matter which two points you take on the curve of \(e^x\), the value of \(f\) at the larger x-value will always be greater.
Exploring One-to-One Functions
A function is considered one-to-one if each y-value in its range corresponds to exactly one x-value in its domain. This characteristic ensures that the function passes the Horizontal Line Test; a horizontal line will intersect the graph of the function at most once.
For the function \(f(x) = e^x\), since it is strictly increasing, it never takes the same value twice. That means for any two different x-values, the y-values are also different. Therefore, \(f(x) = e^x\) is one-to-one.
  • If \(f(a) = f(b)\), then \(a\) must be equal to \(b\).
  • Every input produces a unique output.
This property is essential in defining the inverse of the function, as each output of the function has a clear and single input.
Understanding Derivatives
The derivative of a function represents the rate at which the function鈥檚 value changes at a given point. It gives you the slope of the tangent line at any point on the function's graph.
For the exponential function \(f(x) = e^x\), the derivative is \(f'(x) = e^x\), which is always positive. This constant positivity indicates that as x increases, the y-value increases exponentially.
Derivatives also help in understanding whether a function is increasing or decreasing:
  • If the derivative of a function is positive over an interval, the function is increasing on that interval.
  • If the derivative is negative, the function is decreasing.
  • If the derivative is zero, it could indicate a local maximum, a local minimum, or a saddle point.
In summary, understanding derivatives is crucial in studying the behavior of functions, including their increasing or decreasing nature and their concavity.

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