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

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Start with the function definition: \(f(x) = x + b\), where \(b\) is a constant.

To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\): \(y = x + b\).

Swap the variables \(x\) and \(y\) to find the inverse: \(x = y + b\).

Solve this equation for \(y\) to express the inverse function: \(y = x - b\).

Understand the graphical relationship: the graph of \(f^{-1}\) is the reflection of the graph of \(f\) across the line \(y = x\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.