Consider this function.
[tex]f(x)=-3 x+3[/tex]
Which graph represents the inverse of function [tex]f[/tex]?
Answer (2)
Swap x and y in the equation y = − 3 x + 3 .
Solve for y to find the inverse function: x = − 3 y + 3 .
y = − 3 x − 3 = − 3 1 x + 1 .
The inverse function is f − 1 ( x ) = − 3 1 x + 1 .
Explanation
Understanding the Problem We are given the function f ( x ) = − 3 x + 3 and we need to find the graph that represents its inverse.
Finding the Inverse Function To find the inverse function, we swap x and y in the equation y = − 3 x + 3 and solve for y .
Swapping Variables Swapping x and y gives us x = − 3 y + 3 .
Solving 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 1 x + 1
Identifying the Graph So, the inverse function is f − 1 ( x ) = − 3 1 x + 1 . This is a linear function with a slope of − 3 1 and a y-intercept of 1. We need to identify the graph that represents this line.
Examples
Understanding inverse functions is crucial in many real-world applications. For example, if f ( x ) represents the cost of producing x items, then f − 1 ( x ) would represent the number of items that can be produced for a cost of x . This concept is widely used in economics, engineering, and computer science to reverse processes and solve for the original input.
The inverse of the function f ( x ) = − 3 x + 3 is f − 1 ( x ) = − 3 1 x + 1 . This means the graph of the inverse is a linear function with a slope of - 3 1 and a y-intercept at 1. To identify the correct graph, look for a line that decreases and crosses the y-axis at the point (0, 1).
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