Math Lesson 16.6.2 - Condition for a Function to have an Inverse

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Welcome to our Math lesson on Condition for a Function to have an Inverse, this is the second 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.

Condition for a Function to have an Inverse

Not all functions have an inverse. For example, f(x) = x² doesn't have an inverse, as when we take the inverse of the possible inverse function f - 1(x), we don't obtain the same function as the original one f(x). This is because

The possible inverse function f - 1(x) is determined in the positive part of real numbers only unlike the original function f(x), which is defined in the whole set of real numbers. When we try to reverse again the function f - 1(x) to obtain f(x), this gives

Again, we switch the places to the variables to obtain the possible original function f(x)

Therefore, the domain has changed from unlimited to the positive part of real numbers. This means we no longer have the same function as the original. This fact indicated that the quadratic function has no inverse.

In general, only injective (one-to-one) functions have an inverse. This is the first condition for a function to have an inverse.

The other condition is caused by the relationship between the domain and range. The following rules apply to the domain and range in inverse functions:

1. The domain of f − 1(x) equals the range of f(x); and
2. The range of f − 1(x) equals the domain of f(x).

This explanation makes it clearer why the parent quadratic function does not have an inverse. Thus, in the original function f(x) = x² both the domain and range have no restrictions in the sense that both x and f(x) can take any value in the set of real numbers. This is not the case for the inverse function f - 1(x) = √x, where the x-values can only be positive or zero. Hence, the domain of f - 1(x) is not the same as the range of f(x), therefore f - 1(x) = √x is not the inverse function of f(x) = x².

Example 2

Which of the following functions have an inverse? Write them where this is possible.

1. f(x) = 3/x
2. g(x) = 5x + 1
3. h(x) = x³
4. i(x) = |x|
5. j(x) = 4 - 2x
6. k(x) = 2 - 3x²

Solution 2

From theory, we know that one of the conditions for a function to have an inverse is to be injective (one-to-one), i.e. for every x-value to have a single corresponding f(x). Thus, we have

1. The domain of this function includes all real numbers except zero. Thus, for x ≠ 0,
   f(x) = 3/x
   x = 3/f(x)
   f^-1 (x) = 3/x
   Thus, we obtained the same function defined in the same domain (x ≠ 0), so the function f(x) has an inverse f - 1(x), which is identical to f(x) itself.
2. There are no restrictions in the domain of g(x). Since this function is injective, it has an inverse. We have
   g(x) = 5x + 1
   5x = g(x) - 1
   x = g(x) - 1/5
   g^-1 (x) = x - 1/5
3. The function h(x) is also injective because for every x-value there is a single g(x) value. You can convince yourself and prove this by applying the horizontal line test in the corresponding graph. Indeed, the parent cubic function lies in the first and third quadrants and it is increasing everywhere. This means it is impossible to obtain the same f(x)-value (i.e. the same y-value) for two different x-values. Given this description, the inverse function of h(x) is found in the following way.
   h(x) = x³
   x = √h(x)
   h^-1 (x) = ∛x
4. The absolute value function is a kind of piecewise function, as discussed in the previous tutorial. Since it can split into two different functions (one for each piece), we obtain two f - 1(x) values for the same x. Indeed,
   i(x) = |x| = x for x ≥ 0-x for x < 0
   The inverse of each individual function is
   i(x) = x
   x = i(x)
   i^-1 (x) = x
   and
   i(x) = -x
   x = -i(x)
   i^-1 (x) = -x
   Thus, for the same x we obtain two y-values, so this is not a function. Therefore, the i function f(x) = |x| has no inverse.
5. The function j(x) = 4 - 2x is injective (one-to-one). Therefore, it has an inverse. We have
   j(x) = 4 - 2x
   2x = 4 - j(x)
   x = 4 - f(x)/2
   j^-1 (x) = 4 - x/2
   j^-1 (x) = 2 - x/2
6. This function is quadratic, so it has no inverse. Indeed,
   k(x) = 2 - 3x²
   3x² = 2 - k(x)
   x² = 2 - k(x)/3
   x = √2 - k(x) /3
   k^-1 (x) = √2 - x/3
   This function is defined for x ≤ 2. Reversing it back to k(x) yields
   [k^-1 (x)]²] = √2 - x/3²
   [k^-1 (x)]² = 2 - x/3
   3[k^-1 (x)]² = |2 - x|
   This function splits in two parts:
   3[k^-1 (x)]² = 2 - x for x ≤ 2
   3[k^-1 (x)]² = -(2 - x) for x > 2
   The first function yields
   x = 2 - 3[k^-1 (x)]²
   and the second function yields
   x = 3[k^-1 (x)]² - 2
   Switching the places of the variables as we did in other cases yields
   k(x) = k^-1 [k^-1 (x)]
   = 2 - 3x²
   and
   k(x) = k^-1 [k^-1 (x)]
   = 3x² - 2
   Thus, since there are two different functions k(x) obtained by the same original function by reversing it twice, this is no longer the original function. Hence, k(x) doesn't have an inverse.

You have reached the end of Math lesson 16.6.2 Condition for a Function to have an Inverse. 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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