Find the inverse of f(x)=x+b (b constant). How is the graph of f^(-1) related to the graph of f?

Verified step by step guidance

1. Start with the function definition: \(f(x) = x + b\), where \(b\) is a constant.
2. To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\): \(y = x + b\).
3. Swap the variables \(x\) and \(y\) to find the inverse: \(x = y + b\).
4. Solve this equation for \(y\) to express the inverse function: \(y = x - b\).
5. Understand the graphical relationship: the graph of \(f^{-1}\) is the reflection of the graph of \(f\) across the line \(y = x\).

Key Concepts

Inverse Function

An inverse function reverses the effect of the original function, mapping outputs back to their inputs. For f(x) = x + b, the inverse function f⁻¹(x) solves for x in terms of y, effectively undoing the addition of b.

Finding the Inverse Algebraically

To find the inverse, replace f(x) with y, swap x and y, then solve for y. For f(x) = x + b, swapping gives x = y + b, so solving for y yields f⁻¹(x) = x - b.

Graphical Relationship Between a Function and Its Inverse

The graph of an inverse function is the reflection of the original function's graph across the line y = x. This symmetry means points (a, b) on f correspond to points (b, a) on f⁻¹.