Show that the graph of the inverse of f(x)=mx+b, where m and b are constants and m≠0, is a line with slope 1/m and y-intercept -b/m.

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Start with the given function: $f(x) = mx + b$, where $m$ and $b$ are constants and $m \neq 0$.

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

Swap the roles of $x$ and $y$ to find the inverse: $x = my + b$.

Solve this equation for $y$ to express the inverse function: subtract $b$ from both sides to get $x - b = my$, then divide both sides by $m$ to isolate $y$: $y = \frac{x - b}{m}$.

Rewrite the inverse function in slope-intercept form: $f^{-1}(x) = \frac{1}{m}x - \frac{b}{m}$. This shows the inverse is a line with slope $\frac{1}{m}$ and $y$-intercept $-\frac{b}{m}$.

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주요 개념

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

Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x. Understanding how to find and interpret inverses is essential for analyzing the relationship between a function and its inverse graphically.

Linear Functions and Their Properties

A linear function has the form f(x) = mx + b, where m is the slope and b is the y-intercept. The graph is a straight line, and the slope indicates the rate of change. Recognizing how slope and intercept affect the line helps in determining the characteristics of its inverse.

Finding the Inverse of a Linear Function

To find the inverse of f(x) = mx + b, solve for x in terms of y, then interchange variables. This process yields f⁻¹(x) = (1/m)x - b/m, showing the inverse is also linear with slope 1/m and y-intercept -b/m. This algebraic manipulation is key to proving the inverse's graph properties.