- An inverse relation is a function if and only if the original function is one-to-one.
- A one-to-one function ensures that each output comes from exactly one input, allowing for a unique inverse.
- The linear function \(f(x) = 2x - 3\) is one-to-one because it is strictly increasing (its slope is \(2
eq 0\)).
- The statement that explains this is: \(\boxed{f(x) \text{ is a one-to-one function.}}\)
Explanation
- Defining the Condition for an Inverse Function
To understand whether the inverse relation of a function \(f(x) = 2x - 3\) is a function, we must first recall the condition for a function to have an inverse. A function \(f\) has an inverse that is itself a function if and only if \(f\) is a one-to-one (injective) function. A function is one-to-one if each distinct input value \(x\) in the domain maps to a unique output value \(y\), and each output value corresponds to exactly one input value.
- Analyzing the Function f(x) = 2x - 3
The function \(f(x) = 2x - 3\) represents a non-horizontal straight line with a slope of 2. Because the slope is non-zero, the function is strictly increasing. In an increasing function, as \(x\) increases, the output \(f(x)\) also increases. This means that for any two distinct values \(x_1\) and \(x_2\), \(f(x_1)
eq f(x_2)\). Thus, the function \(f(x) = 2x - 3\) is a one-to-one function. Because the function is one-to-one, we can successfully define an inverse function \(f^{-1}(x)\) by solving the equation \(y = 2x - 3\) for \(x\), which yields \(x = \frac{y+3}{2}\).
- Evaluating the Options
Now, let's evaluate the given options:
- "The graph of \(f(x)\) passes the vertical line test": This statement simply confirms that the relationship is a function; it does not guarantee that the inverse will be a function.
- "\(f(x)\) is a one-to-one function": This is the necessary and sufficient condition for the inverse relation to be a function.
- "The graph of the inverse of \(f(x)\) passes the horizontal line test": This is not the standard explanation for the property of the original function.
- "\(f(x)\) is not a function": This statement is false.
Consequently, the statement that correctly explains why \(f(x) = 2x - 3\) has an inverse that is a function is that \(f(x)\) is a one-to-one function.
- Conclusion
In conclusion, for any function, the existence of an inverse function depends entirely on whether the original function is one-to-one. Since \(f(x) = 2x - 3\) is a one-to-one function, its inverse relation is indeed a function. Therefore, the correct statement is: \(f(x)\) is a one-to-one function. The final answer is \(\boxed{f(x) \text{ is a one-to-one function.}}\)
Examples
The concept of an inverse function is essential in many everyday tasks, such as unit conversion. Consider the function \(C(F) = \frac{5}{9}(F - 32)\), which converts a temperature from Fahrenheit (\(F\)) to Celsius (\(C\)). Since this function is one-to-one, we can invert it to find the function \(F(C) = \frac{9}{5}C + 32\), which converts a temperature from Celsius back to Fahrenheit. Understanding that the function must be one-to-one allows us to reliably move back and forth between these two scales without ambiguity.