$f(x)=x^2-9$ and $g(x)=x+3$
Which of the following is true?
A. In an inverse relation, coordinates of the ordered pairs are multiplied by 2.
B. The horizontal line test is used to determine if a function's inverse will be a function.
C. For a pair of functions to be inverses, their compositions must be equal to 1.
D. An inverse of a function is always a function.
Answer (1)
An inverse relation swaps the coordinates of ordered pairs, not multiplies them by 2.
The horizontal line test determines if a function's inverse is also a function.
For functions to be inverses, their compositions must equal x , not 1.
An inverse of a function is not always a function; it must pass the vertical line test.
The correct statement is: The horizontal line test is used to determine if a function's inverse will be a function. T h e h or i zo n t a ll in e t es t i s u se d t o d e t er min e i f a f u n c t i o n ′ s in v erse w i ll b e a f u n c t i o n .
Explanation
Analyze the Statements We need to evaluate each statement about inverse functions to determine which one is true.
Evaluate Statement 1 Statement 1: In an inverse relation, coordinates of the ordered pairs are multiplied by 2. This is false. In an inverse relation, the coordinates of the ordered pairs are swapped (x becomes y and y becomes x), not multiplied by 2.
Evaluate Statement 2 Statement 2: The horizontal line test is used to determine if a function's inverse will be a function. This is true. The horizontal line test determines if a function is one-to-one, which means its inverse is a function. If a function passes the horizontal line test, no horizontal line intersects the graph more than once, ensuring that the inverse relation will pass the vertical line test and thus be a function.
Evaluate Statement 3 Statement 3: For a pair of functions to be inverses, their compositions must be equal to 1. This is false. For a pair of functions, f ( x ) and g ( x ) , to be inverses, their compositions must be equal to x , the identity function. That is, f ( g ( x )) = x and g ( f ( x )) = x .
Evaluate Statement 4 Statement 4: An inverse of a function is always a function. This is false. The inverse of a function is not always a function; it must pass the vertical line test. For the inverse to be a function, the original function must pass the horizontal line test.
Conclusion Therefore, the correct statement is: The horizontal line test is used to determine if a function's inverse will be a function.
Examples
Imagine you are designing a lock and key. For the key to perfectly unlock the lock (and vice versa), they must be inverses of each other. The horizontal line test helps ensure that each lock has only one unique key that opens it, making the system reliable. This concept of inverse functions is crucial in cryptography, coding, and many engineering applications where reversing a process is essential.
