Which graph represents the inverse of function [tex]f[/tex]?
[tex]f(x)=-3 x+3[/tex]
Answer (2)
To find the inverse of f ( x ) = − 3 x + 3 , we swap x and y to get x = − 3 y + 3 . Solving for y , we find f − 1 ( x ) = 1 − 3 x . The graph of the inverse function is a line with a slope of − 3 1 and a y-intercept of 1. The final answer is the graph representing this line.
Explanation
Find the inverse function We are given the function f ( x ) = − 3 x + 3 and asked to find the graph that represents its inverse. To do this, we first need to find the equation of the inverse function.
Swap x and y To find the inverse function, we swap x and y in the equation y = − 3 x + 3 and solve for y . This gives us x = − 3 y + 3 .
Solve for y Now we solve for y . Subtract 3 from both sides: x − 3 = − 3 y . Divide both sides by -3: y = − 3 x − 3 = 3 3 − x = 1 − 3 x . So, the inverse function is f − 1 ( x ) = 1 − 3 1 x .
Identify the graph The inverse function is a linear function with a slope of − 3 1 and a y-intercept of 1. We need to identify the graph that has these characteristics.
Examples
Understanding inverse functions is crucial in many real-world applications. For example, if you have a function that converts Celsius to Fahrenheit, the inverse function converts Fahrenheit back to Celsius. Similarly, in cryptography, inverse functions are used to decode messages. Knowing how to find and interpret inverse functions allows you to reverse processes and solve problems in various fields.
To find the inverse of the function f ( x ) = − 3 x + 3 , we swap x and y and solve for y , resulting in f − 1 ( x ) = 1 − 3 x . The graph of this inverse function is a line with a slope of - 3 1 and a y-intercept of 1. Look for the graph that reflects these properties.
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