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Chapter 0: Problem 38

Find the inverse of each function. $$ f(x)=-x+3 $$

Short Answer

Expert verified
The inverse of the function is f^{-1}(x) = -x + 3.

Step by step solution

01

Replace f(x) with y

Start by rewriting the function with 'y' instead of 'f(x)': y = -x + 3
02

Swap x and y

Swap the 'x' and 'y' in the equation to prepare for solving for the inverse function:x = -y + 3
03

Solve for y

Isolate 'y' in the equation to solve for 'y'. Subtract 3 from both sides and then multiply both sides by -1:x - 3 = -yy = - (x - 3)y = -x + 3
04

Replace y with f^{-1}(x)

Write the solution in terms of the inverse function:f^{-1}(x) = -x + 3

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Key Concepts

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

Function Transformation
Function transformation involves modifying a function to produce a new function. When finding the inverse of a function, we use certain transformations to 'undo' the effects of the original function.

To find the inverse of a function:
  • Replace the function notation (like f(x)) with a variable (like y)
  • Swap the variables (exchange x and y)
  • Solve for the new variable (y) in terms of x
These steps effectively 'reverse' the original transformations applied by the function. For example, starting with the function \(f(x) = -x + 3\), we first write it as \(y = -x + 3\) and then swap variables to get \(x = -y + 3\). By solving for y, we find the inverse.
Solving Equations
Solving equations is a fundamental skill in mathematics that helps us find the value of unknown variables. When determining the inverse of a function, solving equations involves isolating the variable we are interested in, usually y.

In the provided solution, we first swapped x and y: \(x = -y + 3\). To solve for y, we isolate y on one side of the equation:
  • Subtract 3 from both sides: \(x - 3 = -y\)
  • Multiply both sides by -1 to solve for y: \(y = -(x - 3)\)
This gives us the inverse function in terms of y. Solving equations like this involves understanding basic algebraic manipulations such as addition, subtraction, multiplication, and division.
Inverse Operations
Inverse operations are actions that 'reverse' the effect of another operation. When finding the inverse of a function, these operations help us revert to the original input from the output.

For example, in the function \(f(x) = -x + 3\), the operations are negating x and adding 3. To reverse this and find the inverse function, we:
  • Swap x and y (original output becomes input)
  • Then solve for y (original input becomes output)
The steps involve:
  • Undoing the addition by subtracting 3
  • Undoing the negation by multiplying by -1
Through these inverse operations, we achieve the function's inverse: \(f^{-1}(x) = -x + 3\).

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Most popular questions from this chapter

Find the domain and range for the function \(f(x)=x^{2}+1\).

Let \(f(x)=3 x^{2}-x, g(x)=4 x-2,\) and \(k(x)=|x+3|\) Find the following. $$ f(-1)+g(-1) $$

Capsize Control The capsize screening value \(C\) is an indicator of a sailboat's suitability for extended offshore sailing. \(C\) is determined by the function \(C=4 D^{-1 / 3} B,\) where \(D\) is the displacement of the boat in pounds and \(B\) is its beam (or width) in feet. Sketch the graph of this function for \(B\) ranging from 0 to \(20 \mathrm{ft}\), assuming that \(D\) is fixed at \(22,800 \mathrm{lb}\). Find \(C\) for the Island Packet \(40,\) which has a displacement of \(22,800 \mathrm{lb}\) and a beam of \(12 \mathrm{~ft}\) \(11 \mathrm{~in} .\) (Island Packet Yachts, www.ipy.com).

Solve each problem. A right triangle has a hypotenuse of \(10 \mathrm{ft}\). If one leg is \(4 \mathrm{ft}\), then what is the length of the other leg?

Graph each function by plotting points and state the domain and range. If you have a graphing calculator, use it to check your results. $$ y=\sqrt{x-10} $$
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