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Chapter 10: Problem 24

Determine whether each function is one-to-one. If it is, find the inverse. $$f(x)=-7$$

Short Answer

Expert verified
The function \( f(x) = -7 \) is not one-to-one and does not have an inverse.

Step by step solution

01

- Understand the Function

Identify the given function, which is a constant function: \[ f(x) = -7 \]. Here, every input value maps to the same output value, -7.
02

- Identify One-to-One Functions

For a function to be one-to-one, each input must map to a unique output, and no output can be repeated. Since all inputs map to -7, this function is not one-to-one because the output value is the same for every input.
03

- Conclude

Determine that a function that is not one-to-one does not have an inverse. Thus, since \( f(x) = -7 \) is not one-to-one, it does not have an inverse function.

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Key Concepts

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

Inverse Functions
Inverse functions reverse the roles of inputs and outputs. If you have a function \( f(x) \), the inverse, \( f^{-1}(x) \), will map the output of \( f \) back to its input. For example, if \( f(a) = b \), then \( f^{-1}(b) = a \). To find an inverse function, follow these steps:
  • Replace \( f(x) \) with \( y \).
  • Interchange x and y.
  • Solve for y.
  • Replace y with \( f^{-1}(x) \).
Not every function has an inverse. Only one-to-one functions, where each input maps to a unique output, have inverses. This is crucial because if there are repeated outputs, it鈥檚 unclear how to reverse the process.
Constant Functions
A constant function has the same output value for every input. It's typically written as \( f(x)=c \), where c is a constant. In our exercise, \( f(x) = -7 \) is such a function. No matter what x you input, the output will always be -7.
Constant functions are straightforward but have key properties:
  • The graph of a constant function is a horizontal line.
  • It is not one-to-one because multiple inputs yield the same output.
  • Because it's not one-to-one, it does not have an inverse function.
These characteristics make constant functions unique and simple to understand. They are a clear example of functions without inverses.
Function Properties
Understanding function properties helps in determining important characteristics like whether a function is one-to-one. Here are some properties to consider:
  • One-to-One Function: Each input maps to a unique output. This means no output value is repeated.
  • Onto Function: Every element in the output range has a corresponding input.
  • Constant Function: Outputs are the same for any given input, as discussed earlier.
  • Inverse Function: A reverse mapping of the original function. Only one-to-one functions have inverses.
Using these properties can provide a deeper understanding of functions and their behaviors. It allows identifying key features such as the possibility of finding an inverse, which is based on the function being one-to-one.

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