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FutureStarrThe inverse of a Function OR
An inverse function is a function that, when written, has its original function on the top and a reduced function and minor function on the bottom. That's right: an inverse is a function with a smaller function on the bottom.
Suppose f assigns each child in a family its birth year. An inverse function would output which child was born in a given year. However, if the family has children born in the same year (for instance, twins or triplets, etc.) then the output cannot be known when the input is the common birth year. As well, if a year is given in which no child was born then a child cannot be named. But if each child was born in a separate year, and if we restrict attention to the three years in which a child was born, then we do have an inverse function. For example,
inverse function, Mathematical function that undoes the effect of another function. For example, the inverse function of the formula that converts Celsius temperature to Fahrenheit temperature is the formula that converts Fahrenheit to Celsius. Applying one formula and then the other yields the original temperature. Inverse procedures are essential to solving equations because they allow mathematical operations to be reversed (e.g. logarithms, the inverses of exponential functions, are used to solve exponential equations). Whenever a mathematical procedure is introduced, one of the most important questions is how to invert it. Thus, for example, the trigonometric functions gave rise to the inverse trigonometric functions. (Source: www.britannica.com)
Substituting in f-1 into f means evaluating f(f-1 (x)). Plugging an f inverse function wherever we used to see an x in the f function makes our equation look like this: f(f-1 (x)) = (2x + 5 - 5) / 2. Right away we see the different steps canceling out. The +5 and the -5 in the numerator undo each other, leaving us here: f(f-1 (x)) = 2x / 2, and then the times 2 and divided by 2 undo to bring us right back to where we started - plain old x. This process actually shows us the mathematical definition of inverse functions: two functions that, when composed with each other, cancel out and get us right back to the x on the inside.
The last thing to cover has to do with specifics regarding what makes a function and therefore what makes its inverse. You'll need to remember that any function can only have one output for each input. One common way to test this is called the vertical line test. By tracing a vertical line across the graph, we can check to see if the relation is indeed a function. If our imaginary vertical line crosses the graph in more than one place, the relation is not a function, but if it passes across the entire graph and only ever touches the graph in one spot, it is. (Source: study.com)