Which function has an inverse that is also a function?
$g(x)=2 x-3$
$k(x)=-9 x^2$
$f(x)=|x+2|$
$w(x)=-20$
Answer (1)
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 , f ( x ) = ∣ x + 2∣ , and w ( x ) = − 20 are not one-to-one.
Therefore, only g ( x ) has an inverse that is also a function: g ( x ) = 2 x − 3 .
Explanation
Understanding the Problem 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).
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. They are also 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 ) = 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 ) = 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 every input maps to the same output. Therefore, w ( x ) does not have an inverse that is also a function.
Conclusion Only g ( x ) = 2 x − 3 is both injective and surjective, meaning it is bijective. Therefore, it is the only function with an inverse that is also a function.
Examples
Imagine you're encoding a secret message. If the encoding function has an inverse, you can decode the message back to its original form. Only functions that are one-to-one and onto (bijective) can be reliably decoded. Linear functions like g(x) are perfect for this because each input has a unique output, ensuring no ambiguity when decoding. This is crucial in cryptography and data transmission, where reversibility is key.
