Inverse Functions

By the definition of a function, we know that each x-value we choose in the domain of the function, will be paired with one and only one y-value. That's the definition of a function.

But some functions -- such as f (x) = 4, or g (x) = x 2 -- pair many different x-values with one y-value. Any horizontal linear function-- such as f (x) = 4 -- pairs a single y-value with every x-value in the domain. Such functions are called many-to-one since many x-values are paired with one y-value.

 A function is one-to-one if each y-value is unique.A function is many-to-one if some or all y-values repeat.

Since we're talking about y-values, and, y-values are defined by horizontal lines parallel to the x-axis; to test if a graph represents a many-to-one function, we check if any horizontal line crosses the curve more than once.

.

.

 If any horizontal line (y-value) crosses the graph of a function more than once the function is many-to-one.If not, the function is one-to-one.

In the diagram:

a, b and c are many-to-one functions.
d, e and f are one-to-one functions. .

.

In many math questions, we need to find the inverse of a given function to solve the problem. For example, in a trig question to solve a right triangle -- if we're given 2 sides, and we're looking for the measure of an angle, we can find the value for one of the angle's trig functions -- and from the inverse of that trig function, we can determine the angle measure.

Problem is, when we invert a function -- its range becomes the domain of the inverse. If the original function is many-to-one, there are range elements paired with more than one domain element -- so the inverse can not be a function.

To make a many-to-one function one-to-one so that its inverse is a function, we restrict or limit the domain to eliminate the repetition of the range elements.

For example: f (x) = x 2 is a many-to-one function symmetric to the y-axis. Every y-value on the left of the y-axis is also on the right of the y-axis. So, to generate the inverse function, we chop the domain in half. We use only the right side of the parabola where the x-values are all greater than or equal to zero. It will, of course be, the function .

Had our function been f (x) = (x – 3) 2 + 5, which is a parabola with vertex at (3, 5), symmetric to the vertical line x = 3, we could restrict the domain to either side of the axis of symmetry.

Our restricted domain will be either -- in either case, it will make the function one-to-one. .

Notation Issue:

The inverse of a function f (x) is denoted f – 1(x) -- which is a dangerous and misleading notation since in algebra, something to the power negative one means the reciprocal of the thing -- not the inverse function. With the trig functions, we've developed names for the inverses. They are Arcsin, Arccos, Arctan etc. but with polynomial, rational, and other types of functions the only notation is f – 1(x).

.

 f – 1(x) denotes the inverse of function f (x)the – 1 is not an exponent!!

.

.

What are we doing when we find an inverse function?

The function rule tells us how to find y [or f (x)] when we know x. The inverse function therefore, must tell us how to find x when we know y.

So what we really want is a statement like x = g (y).

However, mathematical convention dictates that domain elements are called x, and range elements are called y -- and we must respect this convention.

So, to find the rule for a function's inverse -- we solve for x in terms of y and then switch the variable names. Most math teachers and texts these days teach this operation in the reverse order. That is, they switch the x's and y's first and then solve for y. It doesn't matter which way we do it since the results are the same.

Example 1

Find the inverse function rule for the quadratic function f (x) = – (x + 3) 2 – 5.

This is a parabola, opens downward, axis of symmetry at x = – 3, max at y = – 5.

We replace f (x) with y , then solve for x.

y = – (x + 3) 2 – 5 gives us – (y + 5) = (x + 3) 2

now, take square root of both sides so Now we rename the variables, state the restrictions and we're done. Note that the domain and range restrictions are reversed.
The vertex of f (x) is ( – 3, – 5)
whereas f – 1(x) has its vertex at ( – 5, – 3).

.

Example 2

Let's do the same example but the other way round. This time, we'll switch the variable names first, and we'll derive the quadratic from the square root function.

.

Find the inverse function rule for the square root function we switch the variable names square both sides

(x + 3) 2 = – (y + 5)

– (x + 3) 2 = y + 5

y = – (x + 3) 2 – 5

f – 1(x) = – (x + 3) 2 – 5; .

.

.

The graphs of a function and its inverse are symmetric about the line y = x.

If we're given just the graph of a function (no rule of correspondence) and we want to draw the graph of its inverse, we draw the line y = x and use it as a mirror to reflect all the points on the given curve.  Find the inverse function rule and state any restictions on the domain.

1) f (x) = 7x + 3, (linear function)

2) g (x) = – 3(x – 2) 2 + 5, (quadratic function)

3) , (first degree rational function)

4) , (square root function) Find the inverse function rule and state any restictions on the domain.

1) f (x) = 7x + 3, (linear function)

y = 7x + 3 becomes y – 3 = 7x so 2) g (x) = – 3(x – 2) 2 + 5, (quadratic function -- vertex at (2, 5), opens down)

y – 5 = – 3(x – 2) 2 becomes  .

3) , (first degree rational function, VA: x = – 4, HA: y = 3)

y(x + 4) = 3x – 5 which becomes xy + 4y = 3x – 5.
xy3x = – 4y – 5 factors as x(y – 3) = – (4y + 5). 4) , (square root function, vertex at ( – 1, – 7), moves up and right)  .

. (all content © MathRoom Learning Service; 2004 - ).