Show that the graph of the inverse of f(x)=m x+b, where m and b are constants and m ≠0, is a lin

step 1 Okay, so this question is asking about how the Y intercept and the slope change when finding the

Show that the graph of the inverse of f(x)=m x+b, where m...

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Key Concepts

Inverse Function

An inverse function reverses the operation of the original function; for any function f, the inverse function f?¹ satisfies f(f?¹(x)) = x. To find the inverse, you exchange the roles of the dependent and independent variables and solve for the new dependent variable.

Linear Functions

A linear function is typically expressed in the form y = mx + b, where m represents the slope and b represents the y-intercept. Its graph is a straight line, and linear functions are characterized by a constant rate of change.

Graphical Interpretation of Slope and Intercept

The slope of a line determines its steepness by giving the ratio between the vertical and horizontal changes, while the y-intercept is the point where the line crosses the y-axis. When analyzing inverses of linear functions, the slope of the inverse becomes the reciprocal of the original function’s slope, and the intercept shifts accordingly.

Algebraic Manipulation for Inverses

Determining an inverse function involves switching the roles of x and y in the equation and solving for y. This technique is an essential tool in algebra as it systematically exposes the structure of the inverse function and reveals its underlying characteristics such as slope and y-intercept.

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