Verified Answer :

  • Set the given function \(f(x) = 2x - 1\) equal to \(y\), establishing the equation \(y = 2x - 1\).
  • Interchange the variables \(x\) and \(y\) to reflect the inverse nature of the function, resulting in \(x = 2y - 1\).
  • Solve the resulting equation for \(y\) by adding 1 to both sides and dividing by 2 to obtain \(y = \frac{x+1}{2}\).
  • Express the final inverse function as \(\boxed{f^{-1}(x) = \frac{x+1}{2}}\).

Explanation

  1. Analysis of the Function and Objective

We are tasked with finding the inverse function of the linear function \(f(x) = 2x - 1\). An inverse function \(f^{-1}(x)\) essentially reverses the operation performed by \(f(x)\). If \(f(x)\) maps an input \(x\) to an output \(y\), then \(f^{-1}(y)\) must map the output \(y\) back to the original input \(x\). To find this function, we follow a systematic algebraic procedure.

  1. Representing the Function as an Equation

To begin the process of finding the inverse, we denote the function output as \(y\). Thus, we rewrite the function as:
\[y = 2x - 1\]
This equation represents the relationship between the independent variable \(x\) and the dependent variable \(y\).

  1. Interchanging Variables for the Inverse Relation

The defining characteristic of an inverse function is the reversal of the roles of the input and output. To isolate the inverse relationship, we interchange \(x\) and \(y\):
\[x = 2y - 1\]
By swapping the variables, we effectively set up an equation that allows us to determine the original input \(x\) given a specific output value.

  1. Solving for the Inverse Function

Now, we solve for \(y\) to express it in terms of \(x\). First, we add \(1\) to both sides of the equation:
\[x + 1 = 2y\]
Next, we divide both sides by \(2\):
\[y = \frac{x + 1}{2}\]
This result gives us the expression for the inverse function in terms of \(x\).

  1. Concluding the Inverse Function

Finally, we replace \(y\) with the formal notation for the inverse function, \(f^{-1}(x)\). We conclude that:
\[f^{-1}(x) = \frac{x + 1}{2}\]
This function serves as the inverse to the original function \(f(x) = 2x - 1\).

Examples

Inverse functions are essential in real-world scenarios involving conversions. For instance, if you have a function that converts temperature from Celsius to Fahrenheit, \(F(C) = \frac{9}{5}C + 32\), the inverse function \(C(F) = \frac{5}{9}(F - 32)\) is used to convert Fahrenheit back into Celsius. Similarly, in economics, if a function calculates the price of an item after a certain tax is applied, the inverse function can be used to determine the original pre-tax price of the item.

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