Which function has an inverse that is also a function?
[tex]$g(x)=2 x-3$[/tex]
[tex]$k(x)=-9 x^2$[/tex]
[tex]$f(x)=|x+2|$[/tex]
[tex]$w(x)=-20$[/tex]
Answer (2)
The function g ( x ) = 2 x − 3 has an inverse that is also a function because it is both one-to-one and onto. In contrast, the other functions do not meet these criteria. Thus, the correct answer is g ( x ) = 2 x − 3 .
;
A function has an inverse that is also a function if it is bijective (one-to-one and onto).
g ( x ) = 2 x − 3 is a linear function, which is both one-to-one and onto.
k ( x ) = − 9 x 2 and f ( x ) = ∣ x + 2∣ are not one-to-one.
w ( x ) = − 20 is a constant function and not one-to-one. Therefore, the answer is g ( x ) = 2 x − 3 .
Explanation
Understanding the Problem We are given four functions: g ( x ) = 2 x − 3 , k ( x ) = − 9 x 2 , f ( x ) = ∣ x + 2∣ , and w ( x ) = − 20 . We need to determine which of these functions has an inverse that is also a function. A function has an inverse that is also a function if and only if it is bijective, meaning it is both injective (one-to-one) and surjective (onto).
Analyzing Each Function Let's analyze each function:
g ( x ) = 2 x − 3 : This is a linear function. Linear functions are one-to-one because they pass the horizontal line test. Also, linear functions (except for horizontal lines) are onto because their range is all real numbers. Therefore, g ( x ) has an inverse that is also a function.
k ( x ) = − 9 x 2 : This is a quadratic function. Quadratic functions are not one-to-one because they fail the horizontal line test (e.g., k ( 1 ) = − 9 and k ( − 1 ) = − 9 ). Therefore, k ( x ) does not have an inverse that is also a function.
f ( x ) = ∣ x + 2∣ : This is an absolute value function. Absolute value functions are not one-to-one because they fail the horizontal line test (e.g., f ( 0 ) = 2 and f ( − 4 ) = 2 ). Therefore, f ( x ) does not have an inverse that is also a function.
w ( x ) = − 20 : This is a constant function. Constant functions are not one-to-one because they fail the horizontal line test (every x value maps to -20). Therefore, w ( x ) does not have an inverse that is also a function.
Conclusion Based on the analysis above, only the function g ( x ) = 2 x − 3 has an inverse that is also a function.
Examples
Understanding inverse functions is crucial in many real-world applications. For example, in cryptography, inverse functions are used to decode messages. If a function encrypts a message, its inverse decrypts it, allowing secure communication. Similarly, in economics, demand and supply curves can be seen as inverse functions of each other. Knowing one allows you to determine the other, which is essential for market analysis and forecasting.
