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Chapter 1: Problem 450

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

Short Answer

Expert verified
The inverse function is \( f^{-1}(x) = x - 5 \).

Step by step solution

01

Replace Function Notation

Start by letting \( y = f(x) \). So we have the equation \( y = x + 5 \).
02

Swap Variables

Next, swap \(x\) and \(y\) in the equation to start finding the inverse. We get \( x = y + 5 \).
03

Solve for y

Now, solve the equation \( x = y + 5 \) for \( y \). Subtract 5 from both sides to get \( y = x - 5 \).
04

Notation Change to Inverse Function

Finally, replace \( y \) with \( f^{-1}(x) \) to represent the inverse function. Thus, \( f^{-1}(x) = x - 5 \).

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Key Concepts

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

Understanding Function Notation
Function notation is a way to clearly describe the relationship between input and output values within a function. Instead of saying, "a number plus five," function notation uses symbols like \( f(x) = x + 5 \) to define that function. Here, \( f \) is the name of the function, \( x \) is the variable, and \( x + 5 \) is the rule applied to \( x \) to get the output, \( f(x) \).

This format is useful because it shows exactly which variable is being used and what operation is performed on it. When working with inverse functions, this clarity is crucial in determining what to reverse. By setting the notation as \( y = f(x) \), it becomes easier to find the inverse function by rewriting the equation. This process involves changing the output back into the input by undoing the operation described in the function notation.
Steps of Equation Solving
To find the inverse of a function like \( f(x) = x+5 \), one key step involves solving equations.
  • Express the function: Start by expressing it in the form of \( y = x + 5 \). This helps visualize that \( y \) depends on \( x \).
  • Isolate the variable: The goal is to manipulate the equation to solve for one variable. After swapping the variables, you have \( x = y + 5 \). To isolate \( y \), subtract 5 from both sides, leading to \( y = x - 5 \).
Equation solving requires performing inverse operations. In this case, subtraction undoes the original addition, leading us to the formula for the inverse function \( f^{-1}(x) = x - 5 \). The balance and order of operations in equation solving are vital to attaining the correct inverse result.
Simplifying through Variable Swapping
Inverting a function's process involves one particularly crucial step: variable swapping. This step is instrumental in accurately deriving the inverse function. Initially, you might have the function situated as \( y = x + 5 \). Swapping \( x \) and \( y \) involves simply rewriting the equation: replace every instance of \( y \) with \( x \) and vice versa. This leads to \( x = y + 5 \).

Swapping variables effectively flips the roles of input and output. This is important because it reorganizes the function structure, paving the way for solving for the new output variable, \( y \), using the terms derived from \( x \). By doing so, it becomes easier to see how to "undo" the function's original operations, effectively reversing them to find the inverse function, which, in this case, is \( f^{-1}(x) = x - 5 \).

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Most popular questions from this chapter

For the following exercises, use the function values for \(f\) and \(g\) shown in Table 1.23 to evaluate each expression. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {0} & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} \\ \hline f(x) & {7} & {6} & {5} & {8} & {4} & {0} & {2} & {1} & {9} & {3} \\ \hline g(x) & {9} & {5} & {6} & {2} & {1} & {8} & {7} & {3} & {4} & {0} \\ \hline\end{array} $$ $$ f(g(5)) $$

For the following exercises, use the function values for \(f\) and \(g\) shown in Table 1.23 to evaluate each expression. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {0} & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} \\ \hline f(x) & {7} & {6} & {5} & {8} & {4} & {0} & {2} & {1} & {9} & {3} \\ \hline g(x) & {9} & {5} & {6} & {2} & {1} & {8} & {7} & {3} & {4} & {0} \\ \hline\end{array} $$ $$ g(f(3)) $$

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ p(x)=\left(\frac{1}{3} x\right)^{3}-3 $$

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f .\) $$ g(x)=f(-x) $$

For the following exercises, determine whether the function is odd, even, or neither. $$ f(x)=(x-2)^{2} $$
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