Math Lesson 16.6.4 - Graph Method for Finding the Inverse of a Function

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Welcome to our Math lesson on Graph Method for Finding the Inverse of a Function, this is the fourth lesson of our suite of math lessons covering the topic of Inverse Functions, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Graph Method for Finding the Inverse of a Function

Let's use a very simple example to explain how to obtain the graph of the inverse f - 1(x) of a given function f(x). For example, let's choose the function f(x) = 2x. We find the inverse function f - 1(x) as follows:

The following figure shows the graphs of the two functions f(x) and f - 1(x).

It is easy to see that the two functions are symmetrical to each other in respect to the line y = x, which acts as a symmetry line. Look at the figure below.

This property of the inverse function graphs, i.e. they are symmetrical to f(x) in respect to the line y = x is used for identifying the inverse of a function f(x) when its graph is given. Let's see an example in this regard.

Example 4

Use the symmetry to find the inverse of the function f(x) in the figure.

Solution 4

First, we have to plot the symmetry line y = x and see how close from it the graph of f(x) is at different points. Look at the figure below.

Let's focus on two points: at the intercept of the two lines [it is at (-4, -4)] and at the y-intercept of f(x), which is at (0, 4). The symmetry rules require that f - 1(x) have a common intercept with the two other lines, i.e. at (-4, -4). Moreover, the y-intercept of f(x) will correspond to the x-intercept of f - 1(x), i.e. at (4, 0). Therefore, we will obtain a new line for f - 1(x) passing through A(-4, -4) and B(4, 0), as shown in the figure below.

Now, let's find the equation of f - 1(x). Since two points of f - 1(x) are already known, we use the formula

to calculate its gradient. Substituting the known values yields,

Thus, the equation of f - 1(x) has the form

where n is a constant to be determined. Substituting the coordinates of any of the known points, for example those of point B(4, 0) in the above function yields

Therefore, the formula of the inverse function f - 1(x) of the function f(x) shown in the figure is

Another method to identify the inverse of a function shown graphically is to first find the formula of the original function from the figure and then use any of the analytical methods described earlier in this tutorial for identifying the formula of the inverse function. This avoids the necessity to use the symmetry line y = x in the process. Let's consider the example below.

Example 5

Find the inverse of the functions f(x), g(x) and h(x) shown in the figure below by identifying the formulas of these functions first.

Solution 5

For all functions shown in the figure, we first identify two points in the graph; then we find the gradient using the known formula in the previous example; and finally, we find the constant n. The next step involves finding the formula of the corresponding inverse function by using any of the two analytical methods described in theory.

1. We can identify two points in f(x); for example, (0, 1) and (3, -5). The gradient of this function is
   m = y₂ - y₁/x₂ - x₁
   = -5 - 1/3 - 0)
   = -6/3
   = -2
   Thus, in the formula
   f(x) = -2x + n
   we have to identify the constant n by substituting in it any known point of the graph, for example (0, 1). Thus,
   1 = -2 ∙ 0 + n
   n = 1
   Hence, the formula of the function f(x) is
   f(x) = -2x + 1
   Now, let's find the inverse function f - 1(x). We have
   f(x) = -2x + 1
   -2x = f(x) - 1
   x = -[f(x) - 1]/2
   f^-1 (x) = -x + 1/2
   f^-1 (x) = -x/2 + 1/2
2. We can identify two points in g(x); for example, (0, -3) and (1, 0). The gradient of this function is
   m = y₂ - y₁/x₂ - x₁
   = 0 - (-3)/1 - 0
   = 3/1
   = 3
   Thus, in the formula
   g(x) = 3x + n
   we have to identify the constant n by substituting in it any known point of the graph, for example (1, 0). Thus,
   0 = 3 ∙ 1 + n
   n = -3
   Hence, the formula of the function g(x) is
   g(x) = 3x - 3
   Now, let's find the inverse function g - 1(x). We have
   g(x) = 3x - 3
   3x = g(x) + 3
   x = g(x) + 3/3
   g^-1 (x) = x + 3/3
   g^-1 (x) = x/3 + 1
3. We can identify two points in h(x); for example, (0, -5) and (3, 1). The gradient of this function is
   m = y₂ - y₁/x₂ - x₁
   = 1 - (-5)/3 - 0
   = 6/3
   = 2
   Thus, in the formula
   h(x) = 2x + n
   we have to identify the constant n by substituting in it any known point of the graph, for example (3, 1). Thus,
   1 = 2 ∙ 3 + n
   1 = 6 + n
   n = -5
   Hence, the formula of the function h(x) is
   h(x) = 2x - 5
   Now, let's find the inverse function h - 1(x). We have
   h(x) = 2x - 5
   2x = h(x) + 5
   x = h(x) + 5/2
   h^-1 (x) = x + 5/2
   h^-1 (x) = 1/2 x + 5/2

You have reached the end of Math lesson 16.6.4 Graph Method for Finding the Inverse of a Function. There are 6 lessons in this physics tutorial covering Inverse Functions, you can access all the lessons from this tutorial below.

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