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Chapter 3: Problem 33

For the following exercises, evaluate or solve, assuming that the function \(f\) is one-to-one. If \(f(6)=7, \mathrm{fi} \mathrm{d} f^{-1}(7)\).

Short Answer

Expert verified
\( f^{-1}(7) = 6 \).

Step by step solution

01

Understanding the Problem

We are given a function \( f \) which is one-to-one, meaning each output is paired with exactly one input. The problem asks us to find the inverse function value \( f^{-1}(7) \), given that \( f(6) = 7 \). An inverse function reverses the roles of inputs and outputs.
02

Identify the Input-Output Pair

Since \( f(6) = 7 \), it means when the input is 6, the output is 7. In the case of an inverse function \( f^{-1} \), we swap the input and output, so \( f^{-1}(7) = 6 \).
03

Verify the Inverse Property

To ensure the inverse is correct, apply the property \( f(f^{-1}(b)) = b \) for any value \( b \). Here, \( f(f^{-1}(7)) = f(6) = 7 \), confirming the inverse function has been correctly identified.

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Key Concepts

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

One-to-One Function
A one-to-one function, also known as an injective function, is a type of function where every output corresponds to exactly one unique input.
This means no two different inputs can have the same output.

To visualize this, consider a simple mapping or pairing system:
  • If you have a set of students and a set of lockers, a one-to-one function would ensure that each student gets their own locker, and no locker is shared nor left out.
  • In mathematical terms, for a function \( f(x) \), if \( f(a) = f(b) \), then it must be true that \( a = b \).
This characteristic is crucial when dealing with inverse functions, because they can only exist if the function is one-to-one.
This is why, in the given exercise, knowing that \( f \) is one-to-one assures us that the inverse function \( f^{-1} \) will be valid.
Function Evaluation
Function evaluation is the process of determining the output of a function when a specific input is given.
In the case of the function \( f(x) \), evaluating the function involves plugging in the input value in place of \( x \) and calculating the corresponding output.

For example:
  • If \( f(6) = 7 \), then upon evaluating \( f \) at \( x = 6 \) we find that the output is 7.
Function evaluation is essentially the reverse when it involves inverse functions. Instead of finding an output, you find the input that gives a certain output.
This is evident in the exercise when we were tasked with finding \( f^{-1}(7) \), which evaluates the function \( f^{-1} \) to give us the input that returned the output of 7 in the original function.
Input-Output Pairs
The concept of input-output pairs is central to understanding how functions operate.
Each pair consists of an input \( x \) and a corresponding output \( f(x) \).

In a one-to-one function, each input-output pair is unique:
  • This guarantees that every input has one specific output, and this pairing does not repeat with different inputs having the same output.
The exercise illustrates this through the given pair \( (6, 7) \):
  • Where 6 is the input and 7 is the output.
  • When finding the inverse function pairs, these roles are reversed, forming the pair \( (7, 6) \), with 7 as the input and 6 as the output.
This swapping of roles when considering inverses is a fundamental aspect of understanding how inverse functions work.
Inverse Property
The inverse property is a defining feature of inverse functions.
It states that for a function \( f \) and its inverse \( f^{-1} \), the following must hold true:
  • \( f(f^{-1}(x)) = x \)
  • \( f^{-1}(f(x)) = x \)
These properties confirm that applying a function and then its inverse returns you to your original input.

In the exercise, the inverse property was used to confirm that \( f^{-1}(7) = 6 \):
  • By checking that \( f(f^{-1}(7)) = f(6) = 7 \), we verify that the correct inverse value was found.
  • This property is crucial in ensuring that inverses accurately reverse the original function's operations.
Understanding and correctly applying the inverse property can greatly aid in solving problems involving one-to-one and inverse functions.

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Most popular questions from this chapter

For the following exercises, use the functions \(f(x)=2 x^{2}+1\) and \(g(x)=3 x+5\) to evaluate or find the composite function as indicated. $$f(g(x))$$

For the following exercises, use the function values for \(f\) and \(g\) shown in Table 4 to evaluate the expressions. $$\begin{array}{||cc|c|c|c|c|c|c|c|} \hline {x} & {-3} & {-2} & {-1} & {0} & {1} & {2} & {3} \\ \hline {f(x)} & {11} & {9} & {7} & {5} & {3} & {1} & {-1} \\\ \hline {g(x)} & {-8} & {-3} & {0} & {1} & {0} & {-3} & {-8}\\\ \hline \end{array}$$ $$(f \circ f)(3)$$

Find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). $$ h(x)=\sqrt{\frac{2 x-1}{3 x+4}} $$

Use the function values for \(f\) and \(g\) shown in Table 4 to evaluate the expressions. $$ \begin{array}{|c|r|r|r|r|r|r|r|} \hline \boldsymbol{x} & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 11 & 9 & 7 & 5 & 3 & 1 & -1 \\ \hline \boldsymbol{g}(\boldsymbol{x}) & -8 & -3 & 0 & 1 & 0 & -3 & -8 \\ \hline \end{array} $$ $$ (g \circ g)(1) $$

For the following exercises, find \(f^{-1}(x)\) for each function. $$ f(x)=\frac{2 x+3}{5 x+4} $$
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