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Chapter 9: Problem 69

Find the inverse of each function. $$f(x)=2 x-1$$

Short Answer

Expert verified
The inverse function is \( f^{-1}(x) = \frac{x + 1}{2} \).

Step by step solution

01

- Write the function

Start with the given function: \[ f(x) = 2x - 1 \]
02

- Replace f(x) with y

Rewrite the function using y instead of f(x): \[ y = 2x - 1 \]
03

- Swap x and y

To find the inverse function, interchange x and y: \[ x = 2y - 1 \]
04

- Solve for y

Isolate y to express the function in terms of x: First, add 1 to both sides: \[ x + 1 = 2y \] Then, divide by 2: \[ y = \frac{x + 1}{2} \]
05

- Write the inverse function

Replace y with \( f^{-1}(x) \) to denote the inverse function: \[ f^{-1}(x) = \frac{x + 1}{2} \]

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Key Concepts

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

finding inverse functions
To find an inverse function, you essentially reverse the roles of the input and output. This helps in determining a function that 'undoes' the operation of the original. Start by swapping the dependent and independent variables. Then, solve for the new dependent variable to express the relationship in terms of the new independent variable.
For example, if the original function is given as: \[ f(x) = 2x - 1 \] you replace \( f(x) \) with \( y \) and swap \( x \) and \( y \): \[ y = 2x - 1 \] becomes \[ x = 2y - 1 \] Now, isolate \( y \), so you'll get the inverse function:
algebraic manipulation
Algebraic manipulation involves the steps you take to isolate a variable in an equation. By using basic operations like addition, subtraction, multiplication, or division, you can solve for the desired variable.
Starting with the equation from the previous section: \[ x = 2y - 1 \] First, add 1 to both sides: \[ x + 1 = 2y \] Then, divide every term by 2 to isolate \( y \): \[ y = \frac{x + 1}{2} \] These operations allow us to transform the equation into a form where \( y \) is expressed entirely in terms of \( x \). Now, \( y \) represents the output of the inverse function.
function notation
In mathematics, function notation helps in communicating complex relationships in a simple way. Once you have isolated the variable, you use function notation to denote the inverse. For instance, after finding: \[ y = \frac{x + 1}{2} \] you rewrite \( y \) as \( f^{-1}(x) \) (which reads as 鈥榝 inverse of x鈥�): \[ f^{-1}(x) = \frac{x + 1}{2} \] This notation clearly indicates that this new function reverses the operation of the original function. Thus, the series of transformations that led to the new equation is visibly summarized through function notation.

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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. $$(f \circ g)(2)=14$$

Find the proportionality constant and write a formula that expresses the indicated variation. See Example 3. \(A\) varies inversely as \(B,\) and \(A=10\) when \(B=3\).

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