- A function has an inverse if and only if it is injective (one-to-one).
- We test each function for injectivity: \(b(x) = x^2 + 3\), \(d(x) = -9\), and \(p(x) = |x|\) fail the horizontal line test because multiple inputs can produce the same output.
- The linear function \(m(x) = -7x\) is strictly decreasing, meaning each output corresponds to exactly one input.
- Thus, the function with an inverse is \(\boxed{m(x) = -7x}\).
Explanation
- Understanding the Condition for an Inverse Function
A function \(f(x)\) has an inverse that is also a function if and only if it is a bijection from its domain to its codomain. For real-valued functions, this is equivalent to being injective (or one-to-one). A function is injective if it satisfies the condition that \(f(x_1) = f(x_2)\) implies \(x_1 = x_2\) for all \(x_1, x_2\) in the domain. Graphically, this means the function must pass the "horizontal line test," where any horizontal line \(y = c\) intersects the graph of the function at most once.
- Analyzing Each Function
We analyze each given function:
- \(b(x) = x^2 + 3\): This is a parabola opening upward with its vertex at \((0, 3)\). Because it is symmetric about the y-axis, for any value \(k > 3\), the horizontal line \(y = k\) will intersect the graph at two distinct points, \(x = \sqrt{k-3}\) and \(x = -\sqrt{k-3}\). Therefore, this function is not injective.
- \(d(x) = -9\): This is a constant function. A horizontal line \(y = -9\) intersects the graph at every point along the x-axis. Since it intersects more than once, it is not injective.
- \(m(x) = -7x\): This is a linear function. For any two values \(x_1\) and \(x_2\), if \(m(x_1) = m(x_2)\), then \(-7x_1 = -7x_2\). Dividing both sides by \(-7\) gives \(x_1 = x_2\). Thus, this function is injective.
- \(p(x) = |x|\): This function represents a V-shape symmetric about the y-axis. For any positive value \(k\), the horizontal line \(y = k\) will intersect the graph at two points, \(x = k\) and \(x = -k\). Therefore, this function is not injective.
- Conclusion
Based on our analysis, the only function that is injective and therefore has an inverse that is also a function is \(m(x) = -7x\).
Examples
In cryptography, inverse functions are essential for decryption. If you use a simple function to "encrypt" data (like multiplying a message by a secret number), you need the inverse function (dividing by that number) to retrieve the original message. Having a one-to-one function ensures that each encrypted value points uniquely back to the original input, avoiding any ambiguity during the decryption process. If a function is not one-to-one, two different inputs would yield the same output, making it impossible to determine which input produced the result.