Evaluating & Interpreting the Inverse Function of a Linear RelationshipStep 1. Find the inverse of the function. Show
Step 2. Interpret the inverse function by swapping the domain and range of the original function. Evaluating & Interpreting the Inverse Function of a Linear Relationship VocabularyA function is a process that is applied to a given variable. In the linear function, f(x) = ax. The process/function that is applied to x is the process of multiplying by a. An inverse function reverses the process of the original function. The inverse function of the above example would divide x by a, since division is the reverse operation of multiplication. Therefore, {eq}f^{-1}(x) = \frac{x}{a} {/eq}. In context, suppose f(x) represented an income after working x hours for $a per hour. Then {eq}f^{-1}(x) {/eq} would represent the amount of hours worked after making a certain income. Did you notice that we basically just swapped hours worked and income? That is, the function f(x) had values (hours, income) while the inverse function has values (income, hours). It turns out that, if a function f has an inverse, the domain of f is the range of the inverse function {eq}f^{-1} {/eq}; and the range of f is the domain of {eq}f^{-1} {/eq}, the inverse function. Evaluating and Interpreting the Inverse Function of a Linear Relationship: Example 1An internet provider charges an initial equipment fee of $500 and an additional $100 per month for using their satellite internet service. The following function represents this linear relationship: C(x) = 100x + 500 where x is the number of months and C(x) is the combined cost of the equipment and the service. (1). Find, {eq}C^{-1}(x) {/eq} , the inverse function (2). Determine which of the following statements best represents the inverse relationship. (a) The number of months, x, using the satellite internet service for C dollars. (b) The reciprocal of the number of months usage to dollars spent. (c) The number of dollars spent after using the service for x months. (d) The difference between the number of months the service has been used, x, and the total dollars spent, C. Step 1. To find the inverse function Let y = C(x) {eq}y = 100x + 500 {/eq} Swap x and y. x = 100y + 500 Solve for y. {eq}y = \frac{x-500}{100} {/eq} Therefore, {eq}C^{-1}(x) = \frac{x-500}{100} {/eq} is the inverse function for C. Step 2. To determine the correct interpretation, swap the domain and range of the function. For C we have (months, cost). This means for {eq}C^{-1}(x) {/eq}, we would have (cost, months) such that the output is the number of months, x, using the satellite internet service for the input, C, dollars. Therefore, the correct answer is (a). Evaluating and Interpreting the Inverse Function of a Linear Relationship: Example 2Converting temperatures from Celsius to Fahrenheit may be represented with the following linear relationship: {eq}F(x) = \frac{9}{5}x + 32 {/eq} Where x represents Celsius temperatures and F(x) represents Fahrenheit temperatures such that the function values are given by (Celsius, Fahrenheit). (1).Find the inverse, {eq}F^{-1}(x) {/eq} of the original function. (2). Find{eq}F^{-1}(32) {/eq} and interpret it. Step 1. To find the inverse function: Let y = F(x) {eq}y = \frac{9}{5}x + 32 {/eq} Swap x and y. {eq}x = \frac{9}{5}y + 32 {/eq} Solve for y. {eq}y = \frac{5}{9}(x - 32) {/eq} Therefore, {eq}F^{-1}(x) = \frac{5}{9}(x - 32) {/eq} is the inverse function for F. Step 2. Find and interpret {eq}F^{-1}(x) {/eq} for x = 32. {eq}F^{-1}(32) = \frac{5}{9}(32 - 32) = 0 {/eq} The inverse function values are represented by (Fahrenheit, Celsius) such that when the Fahrenheit temperature is 32 degrees, the Celsius temperature is 0 degrees. Get access to thousands of practice questions and explanations! If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.
How do you find the inverse of a linear function?The method for finding a function's inverse can be summarized in two steps: Step 1: rearrange the expression y=f(x) to make x the subject. By the end of this you sould have an expression looking like x=f(y). Step 2: swap x and y in the expression obtained at the end of Step 1, the expression obtained is y=f−1(x).
Is the inverse of a linear function linear?Theorem. The inverse of a linear bijection is linear.
Is the inverse of a linear function always a function?The inverse of a function may not always be a function! The original function must be a one-to-one function to guarantee that its inverse will also be a function. A function is a one-to-one function if and only if each second element corresponds to one and only one first element. (Each x and y value is used only once.)
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Inverses of linear functions common core algebra 2 homework
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