Question

Group Activity Inverse Functions Let \(y=f(x)=m x+b\) \(m \neq 0\) (a) Writing to Learn Give a convincing argument that \(f\) is a one-to-one function. (b) Find a formula for the inverse of \(f .\) How are the slopes of \(f\) and \(f^{-1}\) related? (c) If the graphs of two functions are parallel lines with a nonzero slope, what can you say about the graphs of the inverses of the functions? (d) If the graphs of two functions are perpendicular lines with a nonzero slope, what can you say about the graphs of the inverses of the functions? .

Short answer

The function \(f(x) = mx + b\) is one-to-one. Its inverse is \(f^{-1}(x) = (x - b)/m\). The slopes of a function and its inverse are reciprocals. The graphs of the inverses of two parallel functions are also parallel, and the graphs of the inverses of two perpendicular functions are also perpendicular.

Step-by-step solution

Understanding one-to-one functions

A function is one-to-one if no two different inputs give the same output. For the linear function \(y = mx + b\), where \(m \neq 0\), for any two different inputs \(x_1\) and \(x_2\), the outputs \(y_1 = mx_1 + b\) and \(y_2 = mx_2 + b\) will be different because \(m\) is not zero. Therefore, \(f\) is a one-to-one function.

Finding the inverse of the function

To find the inverse of the function \(y = mx + b\), we interchange the roles of \(x\) and \(y\). So, we start with \(x = my + b\) and solve for \(y\) to find the inverse function. This gives \(y = (x-b)/m\). Hence the inverse function \(f^{-1}(x) = (x - b)/m\). The slopes of the linear function and its inverse are reciprocals of each other.

Understanding the relation of graphs of inverses with parallel lines

If two functions are parallel, their slopes are identical. However, the slopes of their inverses are the reciprocals of the original slopes, hence they will also be the same. This means the inverses of two parallel lines will also be parallel.

Understanding the relation of graphs of inverses with perpendicular lines

If two lines are perpendicular, the product of their slopes is -1. However, the inverses of their slopes (i.e., the slopes of the inverses of the functions) will be the negatives of their reciprocals. Therefore, the graphs of the inverses of two perpendicular lines will also be perpendicular.