Example: Finding the Inverse of the Function
To find the inverse of the one-to-one function , follow the standard algebraic procedure. Step 1: Substitute for , yielding . Step 2: Interchange the variables and , which results in . Step 3: Solve for by subtracting from both sides to get , and then dividing by , resulting in . Step 4: Substitute the inverse notation for , yielding . Step 5: Verify the functions are inverses by evaluating their compositions. Checking gives . Checking gives . Since both compositions equal , the inverse function is confirmed.
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Ch.10 Exponential and Logarithmic Functions - Intermediate Algebra @ OpenStax
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Example: Finding the Inverse of the Function
Example: Finding the Inverse of the Function
Example: Finding the Inverse of the Function
In professional roles such as logistics, inventory management, or financial analysis, you often need to reverse a mathematical formula to find an input value from a known output. This involves finding an inverse function. Arrange the following steps in the correct algebraic order to find the inverse of a one-to-one function .
A business analyst uses a one-to-one function to model the relationship between advertising spend and total revenue. To determine the advertising spend required to reach a specific revenue goal, the analyst must find the inverse function. When following the standard five-step algebraic process to find this inverse, which step specifically represents the act of swapping the function's domain and range?
In technical roles such as financial modeling or data science, accuracy is confirmed through specific verification steps. In the standard five-step algebraic process for finding the inverse of a one-to-one function , the final step is to verify the result by checking that the composition of the function and its inverse, , simplifies to the identity value .
In technical careers such as engineering or data analysis, reversing a mathematical process is essential for troubleshooting and forecasting. Match each step of the standard five-step algebraic process for finding the inverse of a one-to-one function with its specific action.
Reversing a Revenue Model
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A manufacturing company uses the function to calculate the total cost of production, where represents the number of units produced. To determine a formula for the number of units based on a known cost, the production manager needs to find the inverse function. Arrange the following steps in the correct order to find the inverse of .
A billing specialist is reviewing a new automated pricing model and determines that the inverse of the pricing function is . According to the standard algebraic procedure, which specific step must they recall and perform to verify that this inverse function is completely accurate?
A billing specialist at a local utility company uses the function to determine a customer's total bill, where represents the hours of service. To create a lookup tool that finds the number of hours from a known total bill , the specialist needs to document the algebraic steps to find the inverse function. Match each procedural step of the inversion process with the specific equation it produces for this function.
Inverse Function for Expense Modeling
A logistics coordinator for a local delivery service uses the linear function to calculate the total shipping cost based on the package weight in pounds, . To create a tool that calculates weight from cost, the coordinator follows the standard algebraic procedure to find the inverse function. True or False: According to this procedure, the step of interchanging the variables and for this function results in the equation .