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Chapter 9: Problem 68

Find the inverse of each function. $$f(x)=x-1$$

Short Answer

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

Step by step solution

01

Understand the Function

The given function is \(f(x) = x-1\). The first step in finding the inverse of the function is to understand that we need to solve for \(x\) in terms of \(y\) where \(y = f(x)\).
02

Replace f(x) with y

Instead of \(f(x)\), write \(y\). This gives us the equation \(y = x-1\).
03

Solve for x

To find the inverse function, solve the equation \(y = x-1\) for \(x\). Add 1 to both sides of the equation, which gives \(x = y + 1\).
04

Write the Inverse Function

Now, write the inverse function by replacing \(y\) with \(x\). Therefore, the inverse function is \(f^{-1}(x) = x + 1\).

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Key Concepts

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

inverse functions
Inverse functions are functions that 'reverse' the effect of the original function. If you have a function that takes you from point A to point B, the inverse function takes you back from point B to point A. When we have a function \(f\) and its inverse \(f^{-1}\), they satisfy the condition \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). Understanding and finding inverse functions can help solve equations where we need to 'undo' the effects of a given function.
algebraic manipulation
Algebraic manipulation involves rearranging and solving equations using operations like addition, subtraction, multiplication, and division. It鈥檚 essential for finding inverse functions. For example, in the exercise given, the function is \(f(x) = x - 1\). To find its inverse, we need to solve for \(x\) in terms of \(y\). By adding 1 to both sides of the equation \(y = x - 1\), we rearrange it to \(x = y + 1\). This step-wise manipulation allows us to ultimately determine the inverse function.
function notation
Function notation is a way to represent functions in an easily readable manner. If \(f(x)\) denotes a function where \(x\) is the input, the result \(f(x)\) is the output. In the given exercise, \(f(x) = x - 1\), and we are asked to find the inverse, denoted as \(f^{-1}(x)\).

Rewriting the function in terms of \(y\), we get \(y = x - 1\). Then, by reversing the roles of \(x\) and \(y\), solving for \(x\) gives us \(f^{-1}(x) = x + 1\). Proper use of function notation keeps our work organized and clear.
solving equations
Solving equations is the process of finding the value(s) of the variable(s) that satisfy an equation. In the context of finding inverse functions, solving equations often involves isolating one variable. For our exercise, we start with \(y = x - 1\).

To isolate, or solve for \(x\), we add 1 to both sides of the equation, getting \(x = y + 1\). Thus, the equation is solved, and the inverse function is \(f^{-1}(x) = x + 1\).

Solving equations requires careful algebraic steps and accurate manipulation to ensure correct solutions.

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Most popular questions from this chapter

Let \(f(x)=x^{2}\) and \(g(x)=x+5 .\) Determine whether each of these statements is true or false. $$(f-g)(0)=5$$

Find the inverse of each function and graph \(f\) and \(f^{-1}\) on the same pair of axes. $$f(x)=x^{2}+3 \text { for } x \geq 0$$

For each pair of functions, find \(\left(f^{-1} \circ f\right)(x)\) $$f(x)=3 x-9 \text { and } f^{-1}(x)=\frac{1}{3} x+3$$

Let \(f(x)=x^{2}\) and \(g(x)=x+5 .\) Determine whether each of these statements is true or false. $$(f \cdot g)(3)=72$$

Classify each function as either a linear, constant, quadratic, square-root, or absolute value function. $$f(x)=-3$$
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