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Magazine Chapter 2: Problem 24
give an example of: An invertible function that is not differentiable at \(x=0\)
Short Answer
Expert verified
An example is a piecewise function like \(f(x) = |x|\) with domain restrictions.
Step by step solution
01
Understanding Invertible Functions
An invertible function is one where for every output, there is a unique input. This means the function must be one-to-one (injective) and onto (surjective). In simple terms, you should be able to recover every input from the output.
02
Choosing a Non-Differentiable Function
Commonly, functions like the absolute value function, which are not differentiable at certain points, come to mind. The function \(f(x) = |x|\) is not differentiable at \(x = 0\).
03
Ensuring Invertibility
The function \(f(x) = |x|\) isn't invertible because it's not one-to-one. However, by restricting the domain, for example, \(f(x) = x\) for \(x \ge 0\) or \(f(x) = -x\) for \(x \le 0\), we can make it invertible. Let's consider \(f(x) = x\) for \(x \ge 0\) which is invertible.
04
Define the Invertible Function
We define the function as \(f(x) = \begin{cases} x, & \text{if } x \ge 0 \ -x, & \text{if } x < 0 \end{cases}\). Its inverse is given by \(f^{-1}(x) = \begin{cases} x, & \text{if } x \ge 0 \ -x, & \text{if } x < 0 \end{cases}\).
05
Confirm Non-Differentiability at x=0
The function \(f(x)\) as defined has a sharp point at \(x=0\), which makes it non-differentiable at that point. The left-hand derivative \(\lim_{h \to 0^-} \frac{f(h) - f(0)}{h} = -1\) and the right-hand derivative \(\lim_{h \to 0^+} \frac{f(h) - f(0)}{h} = 1\) are not equal.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Non-Differentiable Function
A function is called non-differentiable at a certain point if the derivative doesn't exist there. This often happens when the graph of the function has a sharp corner or cusp at that point. A classic example is the absolute value function, \( f(x) = |x| \), which is not differentiable at \( x = 0 \). Here, the graph abruptly changes direction, forming a sharp point at zero.
Non-differentiability can result from:
Non-differentiability can result from:
- Discontinuities in the function
- Sharp corners or cusps
- Vertical tangent lines, where the slope approaches infinity
Absolute Value Function
The absolute value function, denoted as \( f(x) = |x| \), outputs the non-negative value of \( x \). This means it transforms all inputs into their positive counterparts, or leaves them unchanged if they're already positive.
This function is visualized as a V-shaped graph:
This function is visualized as a V-shaped graph:
- For \( x \geq 0 \), the function behaves like \( f(x) = x \)
- For \( x < 0 \), it behaves like \( f(x) = -x \)
Injective and Surjective Functions
For a function to be invertible, it must be both injective and surjective. Let's break these terms down:
- Injective (One-to-One): Each element of the domain is mapped to a unique element in the codomain. Essentially, no two different inputs produce the same output.
- Surjective (Onto): Every element in the codomain is an output of the function. This means the range of the function is exactly the codomain.
Domain Restriction
Domain restriction is a technique used to make a non-invertible function invertible. Some functions, like the absolute value function, aren't naturally one-to-one over their entire domain. By limiting the domain, we transform the function into an invertible one.
For instance:
For instance:
- The function \( f(x) = |x| \) isn't injective over \( \mathbb{R} \) as both positive and negative inputs result in the same output.
- Restricting the domain to \( x \geq 0 \) changes the function to simply \( f(x) = x \), making it invertible.
