Assignment 51
The concept of a function inverse is important. A characterization (not a definition) of a function is that is is a collection of ordered pairs. The inverse of a function is a collection of ordered pairs obtained simply by interchanging the x and y values. As a simple example, if the side of a cube is x inches, then the function f(x) = y = x^{3} would provide the volume (in cubic inches) of the cube. For instance, if a cube has a side of 3.8 inches, then y = f(3.8) = 3.8^{3} = 54.872 (cubic inches) is the volume of the cube. Now, note that
Now think! We have a rule that simply says that if we are given the volume of a cube, we can find the side simply by taking the cube root of the volume. Remember, our original rule was f(x) = x^{3}. OK, the inverse rule is f^{-1}(x) = x^{1/3}. The variable x is a dummy variable. As long as you know what it represents, and you have the correct formula, everything is totally logical. It is the rule that is important, and the rule can be expressed in terms of any variable you choose. So, if x is the length of a side of a cube, and if x = 7.34 inches, then f(x) = x^{3} = (7.34)^{3} = 395.45 (cu. in.) is the volume of the cube. And, if x is the volume of a cube, and if x = 327.76 cu. inches, then f^{-1}(x) = x^{1/3} = (327.76)^{1/3} = 6.895 (in.) is the length of a side of the cube. =============== OK, here is a financial example. If $1,000 is invested for for 25 years at interest rate x, then f(x) = y = $1000(1+x)^{25} will give you the accumulation of the investment for an interest rate x. If x = 8%, for example, then f(.08) = $1,000(1.08)^{25} = $6,848.48. Now, if y = 1000(1+x)^{25} ==> (1+x)^{25} = y/1000 ==> 1+x = (y/1000)^{1/25} ==> x = (y/1000)^{1/25} - 1. OK, we now have a rule that will tell us the interest rate earned if someone tells us the accumulation at the end of 25 years. Since our original rule was f, this rule would be f^{-1}, and would be defined by f^{-1}(x) = (x/1000)^{1/25 }- 1. For instance, if x is the amount of the accumulation after 25 years, and if x = $8,834, then f^{-1}(8834) = (8834/1000)^{1/25} - 1 = 0.09105, or approximately 9.1%. MATH IS POWER.
Problem: A function w is a rule that says "take a number, add 12, and then divide what you have by 3." Express this in functional notation, find w^{-1}, and state the rule w^{-1} in words. Solution (with communication): If x represents a number, then the rule w can be written w(x) = (x+12)/3. Hence, w^{-1} is a rule that says "take a number, multiple it by 3, then subtract 12." We can write w^{-1}(x) = 3x - 12. Problem: Use the functions w and w^{-1} as defined in the problem at the right and show that w(w^{-1}(x)) = x and that w^{-1}(w(x)) = x. Solution (with communication): w(w^{-1}(x)) = w(3x-12) = [(3x-12)+12]/3 = 3x/3 = x, and w^{-1}(w(x)) = w^{-1}[(x+12)/3] = 3[(x+12)/3] - 12 = x+12-12 = x. |