INVERSE FUNCTIONS

THE INVERSE of a function undoes the action of that function.



Say, for example, that a function f acts on 5, producing f(5). Then if g is the inverse of f, then g acting on f(5) will bring back 5

g(f(5)) = 5.

Actually, g must do that for all values in the domain of f. And f must do that for all values in the domain of g. Here is the definition:



Functions f(x) and g(x) are inverses of one another if:

f(g(x)) = x and g(f(x)) = x,

for all values in their respective domains.

Example 1. Let f(x) = x + 2, and g(x) = x − 2. Then they are inverses of one another. For g(x), which subtracts 2 from a number, is the inverse of adding 2: f(x).

Formally, according to the definition:

f(g(x)) = f(x − 2) = (x − 2) + 2 = x,

(f adds 2 to its argument), and

g(f(x)) = g(x + 2) = (x + 2) − 2 = x.

(g subtracts 2 from its argument.)

The definition is satisfied.

Problem 1. Let f(x) = x² and g(x) = x½. Show that they are inverses of one another. (The domain of f must be restricted to x 0.)

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f(g(x)) = f(x½) = (x½)² = x,

and

g(f(x)) = g(x²) = (x²)½ = x.



Constructing the inverse

When we have a function y = f(x) -- for example

y = x²

-- then we can often "invert" the equation by solving for x. In this case,



x now appears as a function of y. Therefore on exchanging the variables,



is the inverse function of y = x².

(Taking the square root of a number is the inverse of squaring a number.)

Hence, to construct the inverse of a function y = f(x):

Solve for x, then exchange the variables.

Example 2. What function is the inverse of y = 3x + 4?

Solution. Exchange the sides of the equation, and solve for x:

3x + 4 = y

3x = y − 4

x = y − 4
3 .

Exchange the variables:

y = x − 4
3 .

That function is the inverse of y = 3x + 4.