Final answer:
A function has an inverse if it is one-to-one, which can be tested using the horizontal line test. The inverse is determined by swapping x and y in the original equation and solving for y, resulting in an equation that undoes the original function. Exponential and logarithmic functions commonly demonstrate this inverse relationship.
Step-by-step explanation:
The question asks whether a given function has an inverse, if that inverse is also a function, and requires an explanation of the reasoning behind the answer. To determine if a function has an inverse, we must verify if the original function is one-to-one, meaning each input has only one output and each output has only one input. This is also described as the function passing the horizontal line test.
If a function has an inverse, we can typically find it by switching the x and y values in the original function and then solving for y. This new 'y' becomes the inverse function. For example, if we have y = 2x, the inverse would be x = 2y, which simplifies to y = x/2 after solving for y.
Concepts such as exponential functions, natural log (ln), and their base-10 equivalents show the fundamental relationship between functions and their inverses. The inverse of the exponential function e^x is the natural log ln(x), and similarly, for the base-10, the inverse of 10^x is log10(x).
In algebra, to undo operations, we often use their respective inverses, like using subtraction to undo addition or division to undo multiplication. This is also evident in trigonometry and exponential functions, where functions like sine are undone by arcsine, and e^x is undone by ln(x). This concept of inverse functions is important because they allow us to reverse a process to get back to our original quantity or value.