inverses of functions
The inverse of a function The symmetry of a graph is a mirror image of the curve around a specific line. The inverse of a function is the symmetry of a graph about the line y = x. Take point A (2 ; 5) on the pair of axes. A’ is a mirror image of A about the line y = x, therefore A’ has coordinates (5 ; 2). The question is: Why? How did we know that the coordinates of the point A’would be (5 ; 2)? Well, something very important to remember, since we are working with a reflection in the line y = x, is that the x and y values simple change around. The functioning of this process is a bit more mathematical, however! What it comes down to is the following: Say we have a function y = 2x + 1. When we feed x = 2 into this function, it will give us an output of 5, which is the value of y at point A. When you have the y – value, and need to determine x, you can work the equation from the other side, and start with y = 5, to calculate what x should’ve been to produce y = 5 in the first place. Functions
This may sound obvious to you, but the rearranging of the equation to get x as the output, is exactly what finding the inverse is all about! What you would basically do, is to take the function y = 2x + 1, then swop the x and y around, to get x = 2y + 1, and then rearrange to get y alone again on the left hand side: x 1 1 1 y , or y x . 2 2 2 If you now feed the x - coordinate of the inverse into this new equation, you will get the y – value of 2, which was the original input! This is all way too technical to really matter, since you need to know the input coordinate of the inverse point, and also know that it is the inverse you are working with, and what the equation of the inverse is, just to find what the original input was! All of this when you could’ve just worked with the original equation, but as with a lot of things in Mathematics this is more about teaching you a way of thinking and understanding, than being practical at this stage. What you should remember is that the inverse is simply when x and y swop places to get the equation of the inverse, but then you rearrange to let it be y = again, just for familiarity’s sake. Take a look at the following examples:
Functions
Find the inverse equations of the following functions and write them in the form f -1 (x) = .... (a) f(x) = 3y + 4 Inverse: x = 3y – 4 x + 4 = 3y 1 4 y x 3 3 1 4 f 1 x x 3 3 (b) f(x) = (x – 6)2 + 1 Solution: x = (y – 6) + 1 x – 1 = (y – 6)2 x 1 y 6 6 x 1 y f 1( x) 6 x 1
We will move on now to look at the graphical inverses of functions. To do this we start with some familiar functions and their graphs: Draw the following graphs on separate sets of axes: 1) y = 2x + 3 y = 2x2 y = 2x 2)
Determine the equations of the inverses of the functions above. Functions
3)
Compile tables in order to represent the inverses graphically. x y
–1
0
1
x y
–1
0
2
x y
1 2
1
2
Tip: To work out log21 on your calculator, press: log 1 log 2
The conclusion that we can then draw about inverses is the following: The inverse of a graph is symmetric with regards to the line y=x All straight lines and exponential function inverses will be functions except the inverse of y = k where k is any constant number. (Can you say why?) All inverses of parabolic functions will be relations, not functions, unless the original function has limits. If the domain of g(x) = x2 is restricted to x 0 the graph for g(x) = x2, x 0 is one-to-one and a function. The inverse of g(x) is thus as follows: g 1( x) : x y 2 y x; y0 g 1 ( x ) x ; y 0 or simply g 1( x) x ; y 0 which is one-to-one and a function Functions
This may sound obvious to you, but the rearranging of the equation to get x as the output, is exactly what finding the inverse is all about! What you would basically do, is to take the function y = 2x + 1, then swop the x and y around, to get x = 2y + 1, and then rearrange to get y alone again on the left hand side: x 1 1 1 y , or y x . 2 2 2 If you now feed the x - coordinate of the inverse into this new equation, you will get the y – value of 2, which was the original input! This is all way too technical to really matter, since you need to know the input coordinate of the inverse point, and also know that it is the inverse you are working with, and what the equation of the inverse is, just to find what the original input was! All of this when you could’ve just worked with the original equation, but as with a lot of things in Mathematics this is more about teaching you a way of thinking and understanding, than being practical at this stage. What you should remember is that the inverse is simply when x and y swop places to get the equation of the inverse, but then you rearrange to let it be y = again, just for familiarity’s sake. Take a look at the following examples:
Functions
Find the inverse equations of the following functions and write them in the form f -1 (x) = .... (a) f(x) = 3y + 4 Inverse: x = 3y – 4 x + 4 = 3y 1 4 y x 3 3 1 4 f 1 x x 3 3 (b) f(x) = (x – 6)2 + 1 Solution: x = (y – 6) + 1 x – 1 = (y – 6)2 x 1 y 6 6 x 1 y f 1( x) 6 x 1
We will move on now to look at the graphical inverses of functions. To do this we start with some familiar functions and their graphs: Draw the following graphs on separate sets of axes: 1) y = 2x + 3 y = 2x2 y = 2x 2)
Determine the equations of the inverses of the functions above. Functions
3)
Compile tables in order to represent the inverses graphically. x y
–1
0
1
x y
–1
0
2
x y
1 2
1
2
Tip: To work out log21 on your calculator, press: log 1 log 2
The conclusion that we can then draw about inverses is the following: The inverse of a graph is symmetric with regards to the line y=x All straight lines and exponential function inverses will be functions except the inverse of y = k where k is any constant number. (Can you say why?) All inverses of parabolic functions will be relations, not functions, unless the original function has limits. If the domain of g(x) = x2 is restricted to x 0 the graph for g(x) = x2, x 0 is one-to-one and a function. The inverse of g(x) is thus as follows: g 1( x) : x y 2 y x; y0 g 1 ( x ) x ; y 0 or simply g 1( x) x ; y 0 which is one-to-one and a function Functions
