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)
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, so they do not have inverses that are functions.
w ( x ) = − 20 is a constant function and is not one-to-one, so it does not have an inverse that is a function. The answer is g ( x ) = 2 x − 3 .
Explanation
Problem Analysis We are given four functions and need to determine which one has an inverse that is also a function. A function has an inverse that is also a function if it is bijective, meaning it is both injective (one-to-one) and surjective (onto). Let's analyze each function.
Analyzing g(x) Consider 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 (with non-zero slope) are onto because their range is all real numbers. Therefore, g ( x ) has an inverse that is also a function.
Analyzing k(x) Consider 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.
Analyzing f(x) Consider 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.
Analyzing w(x) Consider w ( x ) = − 20 . This is a constant function. Constant functions are not one-to-one because they fail the horizontal line test (every x maps to the same y value). Therefore, w ( x ) does not have an inverse that is also a function.
Conclusion Only 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 instance, in cryptography, inverse functions are used to decode messages. If a function encodes a message, its inverse decodes 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 decision-making.
Only the function g ( x ) = 2 x − 3 has an inverse that is also a function, as it is both one-to-one and onto. The other functions analyzed do not meet these criteria. Therefore, the chosen option is g ( x ) = 2 x − 3 .
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