Question

Suppose \(g\) is a real function on \(R^{1}\), with bounded derivative (say \(\left|g^{\prime}\right| \leq M\) ). Fix \(\varepsilon>0\), and define \(f(x)=x+\varepsilon g(x)\), Prove that \(f\) is one-to-one if \(\varepsilon\) is small enough. (A set of admissible values of \(\varepsilon\) can be determined which depends only on \(M\).

Short answer

Question: Prove that for any function \(g:R^{1} \rightarrow R^{1}\) with bounded derivative \(\left|g^{\prime}\right| \leq M\), there exists a set of admissible values for \(\varepsilon\), which depends only on \(M\), such that the function \(f(x)=x+\varepsilon g(x)\) is one-to-one (injective) for these values of \(\varepsilon\). Answer: Choose \(\varepsilon\) such that \(\varepsilon < \frac{1}{M}\).

Step-by-step solution

Use the Mean Value Theorem to find a relation for \(f'(x)\) and apply bound on \(g'(x)\)

We know that \(f ( x ) = x + \varepsilon g ( x )\). First, we need to find its derivative, \(f'(x)\). From the definition of \(f(x)\), we have $$ f'(x) = 1 + \varepsilon g'(x). $$ Now, we apply the given bound on \(g'(x)\): $$ -\varepsilon M \leq \varepsilon g'(x) \leq \varepsilon M. $$ Adding 1 to each of the inequalities will give us a bound for \(f'(x)\), $$ 1-\varepsilon M \leq f'(x) \leq 1+\varepsilon M. $$

Choose an appropriate range of values for \(\varepsilon\)

Our goal is to find a set of values for \(\varepsilon\) such that \(f(x)\) is injective. To achieve injectivity, it is required that \(f'(x) > 0\) for every \(x \in R^{1}\). This condition will ensure that \(f(x)\) is strictly increasing for any \(x\), and thus, making it one-to-one. In order to ensure that \(f'(x) > 0\), we can choose \(\varepsilon\) to satisfy: $$ 1 - \varepsilon M > 0. $$ Rearrange the inequality, we get: $$ \varepsilon < \frac{1}{M}. $$

Conclude that the set of admissible values of \(\varepsilon\) is found

Based on Step 2, we have determined a set of admissible values for \(\varepsilon\), which depends only on \(M\). By choosing \(\varepsilon\) satisfying \(\varepsilon < \frac{1}{M}\), we obtain \(f'(x) > 0\) for every \(x\). Hence, \(f(x)\) is strictly increasing and injective (i.e., one-to-one) when \(\varepsilon\) belongs to this set of admissible values. In conclusion, as long as \(\varepsilon\) is small enough such that \(\varepsilon < \frac{1}{M}\), the function \(f(x)=x+\varepsilon g(x)\) is injective (one-to-one).

Key concepts

Mean Value Theorem

The Mean Value Theorem (MVT) is a foundational concept in calculus that connects the derivative of a function to its overall change over an interval. It states that if a function is continuous on a closed interval \[a, b\] and differentiable on the open interval \(a, b\), then there exists at least one point c in \(a, b\) where the derivative of the function at c is equal to the average rate of change of the function over \[a, b\]. In symbolic form, this is expressed as \(f'(c) = \frac{f(b) - f(a)}{b - a}\).

In the context of the exercise, MVT is applied by considering the function's derivative. By establishing a relationship between the derivative of the function \(f\) and its behavior, the theorem helps in proving injectivity. The derivative represents the slope of the tangent line to the function at any given point. For the function \(f\) to be one-to-one, its slopes must not change signs. Therefore, by ensuring that \(f'(x) > 0\) for all x in the domain, MVT confirms that the slope is always positive, indicating a strictly increasing function which is a required condition for injectivity.

Derivatives

Derivatives are one of the central tools in calculus and provide a measure of how a function changes as its input changes. The derivative of a function at a point is the slope of the tangent line to the function's graph at that point. It is often denoted as \(f'(x)\) or \(\frac{df}{dx}\). A positive derivative suggests that the function is increasing, while a negative derivative indicates that the function is decreasing.

Within our exercise, the derivative \(f'(x) = 1 + \varepsilon g'(x)\) directly influences the behavior of the function \(f\). By controlling the magnitude of \(\varepsilon\), we ensure that the derivative remains positive. If \(\varepsilon\) is small enough, specifically smaller than \(\frac{1}{M}\), where \(M\) bounds \(g'(x)\), \(f'(x)\) is guaranteed to be positive. Consequently, the function \(f\) is monotonically increasing, which is another way of saying that the function moves upwards as x increases. This behavior ensures the function is one-to-one since for each x, there corresponds a unique value of \(f(x)\).

Moreover, the concept of derivatives extends further to cover higher-order derivatives, optimization problems, and the behavior of functions, which all play a role in understanding the intricacies of a function's graph.

Injective Functions

Injective functions, also known as one-to-one functions, play a significant role in understanding how functions behave. A function \(f\) is injective if and only if \(f(x_1) = f(x_2)\) implies \(x_1 = x_2\) for all \(x_1, x_2\) in the domain of \(f\). This means that different inputs into the function produce different outputs; no two distinct inputs map to the same output. Injectivity is a desirable quality in functions that ensures the function is reversible, or in other words, it has an inverse.

In providing a step-by-step solution to guarantee the injectivity of the function \(f\), the exercise utilizes constraints on the derivative of \(f\). By selecting an appropriate value for \(\varepsilon\), which is less than \(\frac{1}{M}\), it is ensured that the derivative is always positive. The critical tie-in with injectivity here is that a function with a constant positive (or negative) derivative is strictly monotonic, and hence, injective. The solution requires a comprehension of both the definition of an injective function and its graphical interpretation, as well as its analytical condition involving the derivative. Through the careful application of these principles, the problem demonstrates the maintenance of injectivity for the function \(f\) with respect to the bounding constant \(M\) and the variable \(\varepsilon\).