# 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

if

• 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

satisfied.

**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

functions:

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 = x^{2}, which fails the test, and y = x^{3}, 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 = x^{2} +2 has no inverse. Notice that

f(2) = 6 = f(−2). So there are two x values that give the same y values, and

x^{2} + 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