Someone want to check my short essay for any mistakes? Devin Varao Mrs. Roberts Algebra II/Trigonometry 10/1/14 The Art of Finding the Inverse to a Function As we draw our attention to the complications of manipulating a function, we may notice a certain complexity in inverting an equation. For those whom are unfamiliar with functions and their operations; a relation can be deemed a function by using a graph through means of a vertical line test. In a written form however, determining that a relation is in fact a function is as simple as finding one and only one x-value for each y-value through the utilization of a table of values. After determining that a relation is a function, we must assess the function in a horizontal line test on a graph (or ensure there is only one y-value for every x-value on a table of values). These tests will ensure that the function is a one-to-one function. In explanation, our procedures for inverting a function are only possible if the relation or line is a one-to-one function. Assuming that one possesses the common knowledge of a function, we shall proceed to the following enumerative procedures of inverting the function: f(x)= -1/3x + 1 To find the inverse of the written-out, one-to-one function “f(x)= -1/3x + 1” we must follow the most important and simplest step of inverting a function; exchanging the x-value for the y-value. This will transfer the equation from “y= -1/3x + 1” to “x= -1/3y + 1”. Next, we must solve for the y-value. To achieve this, we must first multiply 3 to each term on both sides of the equal sign to rid the equation of a fraction. An example of how this will look is: 3x= -y + 3. The remainder of this step is to isolate the y-value: first subtracting “3” from both sides of the equation making it “3x – 3= -y”, then negating both sides since the y-value is multiplied by -1 {e.g. -1( 3x – 3)= -1(-y)} to end up with “y= -3x + 3”. Now we can stick in the inverse notation, f-1(x), neatening and finalizing our equation to “f-1(x)= -3x + 3”. To check our answer we can either graph both “f(x)” and “f-1(x)” of our equation on the same graph and see if they are reflections over “y = x”. Alternatively, we can either find “(f o f-1)(x)” or “(f-1 o f)(x)”. To reflect on what we covered in a simplistic way, we can break it down into short hand steps: Step 1: Switch “x” and “y”, Step 2: Solve for “y”, Step 3: replace” f-1(x)” for the “y”. This process is a relieving way to ease the complexity of inverting an equation. To stress the importance of introducing that these procedures will only work with a one-to-one function, if we were to substitute “x” and “y” in an equation that has multiple y-values, we would end up with multiple x-values which is not a function and would not reflect correctly over our “y = x” mirror. It is also important to note that if we were to skip a step our equation would not accurately depict our desired inverse. In conclusion, one’s confusion may be lulled as long as they first determine that the function is one-to-one, then following each of the necessary steps in determining the inverse of a function, and more importantly double-checking their solution with the possible aforementioned equations.
Posted on: Wed, 01 Oct 2014 01:04:12 +0000
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