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Inverse Functions

Definition 1.1. We will define another special type of composition function,
called the inverse function. A function g is called the inverse function of f

• f (g (x)) = x for all x in the domain of g
• g (f (x)) = x for all x in the domain of f

Be careful to notice that for functions to be inverses, both criteria must be

Example 1.2. Let f (x) = 3x + 1 and  . Notice that f and g are
inverse functions:

Since f (g (x)) = x = g (f (x)) for all x, they must be inverse functions.

Example 1.3. Let and . Notice that f and g are inverse

Since f (g (x)) = g (f (x)) for all x, they must be inverse functions.

Example 1.4. Let , and  What happens to the criteria
for inverses?

Thus, f and g are not inverse function, if we are using all of R as our domain.
The are inverses if we only use [0,∞) as our domain.

Item 1.5. Really, all an inverse function does, is answer the question, “what x
value will give me a y value of --?” Inverses switch the input and the output of a
function, since we HAVE the y value, and WANT the x value, as opposed to the
usual case. Since this switching of input and output is what inverse functions
do, we can use this tool to actually FIND the inverse of a function.

Example 1.6. Find the inverse function for f (x) = 7x + 2. To do so, we will
make use of the same procedure we used to find the range of a function: we will
solve for x in terms of y:

Since inverse functions switch the input and output, change y and x:

is the inverse of f.
Example 1.7. Use the same procedure to find the inverses for ,
and .

Item 1.8. Notice that . Use f (x) = 2x + 1 as a counterexample.

Item 1.9. If there the ordered pair (4, 5) is on the graph of f (x), then the
ordered pair (5, 4) is on the graph of f -1(x), since inverses switch input and
output. If we keep plotting points on f and on f -1, we’ll notice that f -1 is just
a reflection of f over the line y − x. As examples, plot the following graphs:

on the right half-plane, [0,∞)

Definition 1.10. A function is said to be one to one if every y value is the
image of exactly one x value

Item 1.11. If we reflect a vertical line, which will tell us if a graph is a function
(remember the vertical line test gives us a function if the line hits the graph
at only one point), about the line y = x, since that’s what inverses do, we get
a horizontal line. So if an inverse is a function, then reflection across the line
y = x gives us a horizontal line crossing the original function exactly once.
For examples, graph y = x2, which fails the test, and y = x3, which passes the
horizontal line test.

Example 1.12. Show algebraically that y = 7x + 2 has an inverse. Assume
that there are two values, a and b that have the same y value:

So if two numbers give us the same y value, they have to be the same. Thus,
7x + 2 has an inverse.

Example 1.13. Show algebraically that y = x2 +2 has no inverse. Notice that
f(2) = 6 = f(−2). So there are two x values that give the same y values, and
x2 + 2 has no inverse.

Item 1.14. Please do the following problems on pages 238ff. to turn in:

1, 5, 9, 13, 23, 25, 27, 29, 31, 33