The inverse of a function
To solve for an inverse equation:
Let's begin with something simple: finding the inverse of a set of ordered pairs. Let's say that the directions asked us to find the inverse of the following set of ordered pairs:
(2,1) (1,2)
(2,2) (2,2)
(3,1) to find the inverse, switch the x and y values (1,3)
(4,2) (2,4)
(1,3) (3,1)
- switch x and y immediately and then solve for y
Let's begin with something simple: finding the inverse of a set of ordered pairs. Let's say that the directions asked us to find the inverse of the following set of ordered pairs:
(2,1) (1,2)
(2,2) (2,2)
(3,1) to find the inverse, switch the x and y values (1,3)
(4,2) (2,4)
(1,3) (3,1)
The graph to the left shows three different line functions. y = 2x+3 is the inverse of the function y= 1/2(x-3). But, what if we only had the equation y = 2+3 and were asked to find the inverse. We could easily do so by immediately switching x and y and then solving for y:
y = 2x+3
x = 2y+3
x-3 = 2y
(x-3)/2 = y
y = 2x+3
x = 2y+3
x-3 = 2y
(x-3)/2 = y
Let's try a parabola: y = (x+1)² - 3
x=(y+1)²-3
x+3=(y+1)²
SQRT(x+3) = SQRT(y+1)²
SQRT(x+3) = y+1
SQRT(x+3) + 1 = y
Here's a square root: y = SQRT(x+1)-3
x = SQRT(y+1)-3
x+3 = SQRT(y+1)
(x+3)² = y+1
(x+3)² -1 = y
The inverse of a square root is a parabola, the inverse of a parabola is a square root.
Find the inverse of cubic: y = (x+2)³+2
x+2 = (y+2)³
x+2 = (y+2)³
3SQRT(x+2) = 3SQRT(y+2)³
3SQRT(x+2) = y+2
3SQRT(x-2) - 2 = y
x=(y+1)²-3
x+3=(y+1)²
SQRT(x+3) = SQRT(y+1)²
SQRT(x+3) = y+1
SQRT(x+3) + 1 = y
Here's a square root: y = SQRT(x+1)-3
x = SQRT(y+1)-3
x+3 = SQRT(y+1)
(x+3)² = y+1
(x+3)² -1 = y
The inverse of a square root is a parabola, the inverse of a parabola is a square root.
Find the inverse of cubic: y = (x+2)³+2
x+2 = (y+2)³
x+2 = (y+2)³
3SQRT(x+2) = 3SQRT(y+2)³
3SQRT(x+2) = y+2
3SQRT(x-2) - 2 = y
Is the inverse a function?
The same function-finding-rules apply to inverses of equations that apply to all equations. You can always use the vertical line test on a graph to see if an equation is a function: hold your pencil vertically and pass it over the graph, if the pencil is never touching two parts of the graph at once, then it is a function. You could also create a table, map, or list of ordered pairs in order to see if the equation is a function: remember that for an equation to be a function, each x value (input) must have only one y value (output).
Try finding the inverse of the following functions. Click the answer button below to check your work.
f(x)=x-1
g(x)=x²-2x+1
h(x)=x³-1
f(x)=x-1
g(x)=x²-2x+1
h(x)=x³-1