If the function [tex]f(x)=m x+b[/tex] has an inverse function, which statement must be true?
A. [tex]m \neq 0[/tex]
B. [tex]m=0[/tex]
C. [tex]b \neq 0[/tex]
D. [tex]b=0[/tex]
Answer (2)
A linear function f ( x ) = m x + b has an inverse if it is bijective.
The function is a horizontal line if m = 0 , and it does not have an inverse.
If m = 0 , the function is a non-horizontal line and has an inverse.
The condition for f ( x ) to have an inverse is m = 0 , so the answer is m = 0 .
Explanation
Understanding the Problem We are given a linear function f ( x ) = m x + b and asked to determine which condition must be true for the function to have an inverse. A function has an inverse if and only if it is bijective (both one-to-one and onto). For a linear function, this means it must not be a horizontal line.
Analyzing the Slope A linear function f ( x ) = m x + b is a straight line. The slope of this line is m , and the y-intercept is b . If m = 0 , the function becomes f ( x ) = b , which is a horizontal line. A horizontal line fails the horizontal line test, meaning it is not one-to-one and therefore does not have an inverse.
Determining the Condition for an Inverse If m = 0 , the function is a non-horizontal line. Any non-horizontal line is one-to-one and onto, and therefore has an inverse. The value of b (the y-intercept) does not affect whether the function has an inverse.
Conclusion Therefore, the statement that must be true for the function f ( x ) = m x + b to have an inverse is m = 0 .
Examples
In real life, the concept of inverse functions is used in various applications such as converting temperature scales (Celsius to Fahrenheit and vice versa), encoding and decoding messages, and calculating currency exchange rates. For example, if f ( x ) represents converting Celsius to Fahrenheit, then its inverse f − 1 ( x ) would convert Fahrenheit back to Celsius. The existence of an inverse ensures that the conversion can be done in both directions.
The function f ( x ) = m x + b has an inverse if and only if the slope m is not equal to zero. Thus, the correct answer is m = 0 .
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