Show that the graphs of two linear functions f and g are perpendicular if and only if the graph

step 1 We are given the function fx equal to small f plus 3 divided by x minus g x and we need to find

Show that the graphs of two linear functions f and g are...

Key Concepts

Biconditional Statement

The 'if and only if' condition expresses a logical equivalence, meaning that both the forward and reverse implications must hold true. In the context of this problem, it asserts that the graphs of the two given linear functions are perpendicular if and only if the slope of their composite function is -1, establishing a tight relationship between the geometric configuration and the algebraic computation.

Slope of a Composition of Linear Functions

When composing two linear functions, the resulting function is also linear, and its slope is the product of the slopes of the individual functions. This property directly links the algebraic operation of composition with the geometric interpretation of slopes, providing a clear criterion for evaluating conditions like perpendicularity when the composite slope is set to a specific value, such as -1.

Linear Functions

A linear function is a first-degree polynomial function, typically expressed in the form f(x) = mx + b, where m represents the slope of the line and b represents the y-intercept. The slope indicates the rate of change of the function, and it is a fundamental concept in understanding the behavior of linear graphs.

Perpendicular Lines

Two lines in the plane are perpendicular if and only if the product of their slopes is -1. This condition arises from the geometric relationship between the lines, ensuring that the angle between them is 90 degrees, a property that is critical in both analytic geometry and various applications involving orthogonal projections.

Composition of Functions

The composition of functions involves applying one function to the results of another function. For functions f and g, the composition f ? g is defined by (f ? g)(x) = f(g(x)). This concept is significant in understanding the sequential application of transformations and how individual function properties, such as slope, combine during composition.

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