Inverse Functions and One-to-One Functions

An inverse function is the relation formed when the dependent variable is exchanged with the independent variable in a given function.

Given function: 

\[f(x)=2x+1\]

How to find the inverse:

Switch the x and y values in the function.

f^{-1}(x)=2y+1

Then get y on its own

x-1=2y

y=\frac{x-1}{2}

Inverse function:

f^{-1}(x)=\frac{x-1}{2}

Two functions f and g are inverses of each other if

(fog)(x)=(x)

(gof)(x)=(x)

Will all functions have inverses that are also functions?

Not all functions will have inverses that are also functions.

How can you determine if a function's inverse is also a function?

1. Graph the function and its inverse

-The vertical line test will determine if f(x) is a function.

-The horizontal line test of a reflection of f(x) over the identity line will determine if a function's inverse g(x) has an inverse function.

-f(x) passes the vertical line test

-The inverse, g(x), passes the horizontal line test.

-The range of the inverse function is the domain of the original function.

Conclusion: the inverse of the function is also a function

-The inverse f^{-1}(x) is a reflection of the original function f(x) over the identity line y=x.

-The red sketch, 

f^{-1}(x) or g(x),

is the inverse relation of f(x)

.  

-Because the g(x) will not pass the horizontal line test for functions, f(x)

 does not have an inverse function.

Conclusion: The inverse relation exists, but it is NOT a function.

2. Determine algebraically if the function is one-to-one

A function f is One-to-One if, a and b in its domain, f(a) = f(b) that implies that a = b. To have an inverse, no two elements in the domain of f 

may correspond to the same element in the range of f. 

Two special functions that pass Horizontal Line Test

-If f is increasing on its entire domain, f is one-to-one.

-If f is decreasing on its domain, f is one-to-one.

Testing Algebraically

Let a and b be nonnegative real numbers with f(a)=f(b)

\sqrt{a}+1=\sqrt{b}+1

\sqrt{a}=\sqrt{b}

a=b

So f(a)=f(b) implies that 

a=b

Testing Graphically

Graph the function. 

You can see that a horizontal line will intersect the graph once at most

Key Points

• To find inverse
• use horizontal line test to decide whether f has an inverse
• in the equation of f(x), replace f(x) by y
• interchange the roles of x and y, and solve for y
• Replace y by f^{-1}(x) in the new equation
• Verify that f and f^{-1}(x) are inverses of each other by showing that f(f^{-1}(x))=x and f^{-1}(x)=x
• The graph of a function is reflected over the identity line to form the graph of its inverse
• only functions that pass the horizontal line test will inverses that are functions
• a function is one-to-one if f(a)=f(b) implies a=b

Composition of Functions

Composition of Functions Composition is when we take the output of one function and use it as the input for another function. Another way to state this is h(x) = f(g(x)), or:




Note: This is pronounced "f of g of x", not "fog x".

Let's take a look at an example of composition. For this example, we will use:

Composition is defined as:

First, we fill in f and g:

Then, we simplify, getting:

To find the domain of a composite function, we need to first find the domain of g(x). Then, since we are putting the results of g(x) into f(x), we need to find which results of g(x) are in the domain of f(x). If x is in the domain of g(x), and g(x) is in the domain of f(x), then x is in the domain of h(x).

For example:

Filling in f and g, we get:

Since the denominator of g(x) cannot equal 0, x cannot equal 5. Therefore, the domain of g is (-∞, 5) ∪ (5, ∞). But we're not done yet. We cannot have a negative under the square root in f, so we can exclude numbers less than -3 from the domain of f. But, since this is a composite function, g(x) is the domain of f, so g(x) cannot be less than -3. By excluding all x that would make g(x) less than -3, we have the domain of (f ○ g) (x), which is:

Things to Remember:
1. (f ○ g) (x) is pronounced "f of g of x", not "fog x".
2. Always remember to put parentheses in the right places.

Arithmetic Combinations of Functions

Arithmetic Combinations of Functions
We know that functions are a relationship between two unknowns like a ratio is a relationship between two numbers. Relationships between two numbers can be arithmetically combined, ex.

So functions can be combined in a similar way.

What does that mean?:

Arithmetic combinations of functions are the sum, difference, product, or quotient of two functions,  commonly f(x) and g(x), with overlapping domains. That's right they must have overlapping domains (where both functions are real and defined). The operation is just simple arithmetic but be on the lookout for the distributive property and accidentally simplifying parts of expressions.

For each example of the types of arithmetic combinations of functions, we will use: 

Sum of Functions:

Defined as:

The sum of the function f and the function g is equal to the sum of their outputs, respectively.

As for our example, simply add g(x) to f(x):

We could also evaluate an input in this new compound function, which is named (f + g)(x), by inputting a value for x. Let's evaluate (f + g)(1):

If we look at a graphical approach to the sum of functions, it gets a little more complicated. For every value of f(x), we have to add the corresponding value of g(x). 

f(x):

                                                           (f+g)(x):

g(x):

We can see that the graph looks to be exactly what the name (f+g)(x) describes: every y value of g(x) was added to the corresponding value of f(x).

Difference of Functions:

Defined as:

The difference of f(x) and g(x) is equal to the difference of the outputs, respectively.

Example (just subtract g(x) from f(x)):

Now let's evaluate (f - g)(0):

Graphically, (f - g)(x) is the y values of the graph  f(x) minus the y values of g(x), as we can see here:

 f(x):

                                                           (f - g)(x):







g(x):

We can see that all of the graph of f(x) was lowered except where the y value of g(x) is equal to zero.

Product of Functions:

Defined as:

The product of the the functions f(x) and g(x) is equal to the product of their outputs.

In our example, the produce of f(x) and g(x) is found be the following process:

Evaluate (fg)(-1):

In graphing (fg)(x), we would go about it in a similar fashion as before: multiply the output of f(x) by the output of g(x).

f(x):

                                                            (fg)(x)

g(x):

In these graphs, the results are a little extreme but you can still tell that when the outputs of both f(x) and
g(x) are the same sign, the final output is positive. When f(x) and g(x) are opposite signs, the result is negative.

Quotient of Functions:

Defined by:

The quotient of f(x) and g(x) is equal to the output of f(x) divided by the output of g(x).

Following the example above:






And evaluating (f / g)(2):

The graph of (f / g)(x) is has a catch that all others do not. If g(x) is not constant, then the zeroes of g(x) must be excluded from the domain of the function. This is because if g(x) is zero, the whole rational function is undefined. Many problems will ask for a domain along with the equation for (f / g)(x). For our example:

So we expect a vertical asymptote when x equals negative one half. Which is reflected by the graph of
(f / g)(x).

f(x):





                                                           (f / g)(x):



g(x):

General Tips:

1. (f + g), (f - g), (fg), and (f / g) are names not operations, don't let them confuse you!
2. If you are doing subtraction, always put the g(x) in parenthesis and distribute the negative.
3. For (f / g)(x), exclude the values where g(x)=0 because that makes the function undefined.
4. Never simplify expressions. If you have something like the following example, you must factor to simplify.

Section 1.2 Function Definitions

Function Definitions

Functions are relations between variables that have one output for every input. Functions can be categorized as one of six different types, each one having a contradictory type.

Increasing and Decreasing Functions

 A function f is increasing on an interval if, for any andin the interval,

implies

Basically, lesser inputs yield lesser outputs.

An example of an increasing function would be when x=y

A function f is decreasing on an interval if, for any andin the interval,

implies

In this case, lesser inputs yield greater outputs.

When compared to an increasing function, decreasing functions can be the same, but with opposite y values.

Maximum and Minimum Values

A function value f (a) is called a relative maximum of f if there exists an interval
that contains a such that 

 implies

A maximum is global when it is the greatest value in the entire graph, and is local when it is the greatest value within a selected portion of the graph.

A function value f (a) is called a relative minimum of f if there exists an interval

that contains a such that 

 implies 

A minimum value can also be global or local, and both minimum and maximum are used for the y values of a graph.

A function which shows a local maximum and minimum, and its global maximum

Even and Odd Functions

A function f is even if, for each x in the domain of f ,

The most noticeable feature of an even function is that it will always have a line of symmetry at x=0. The y values will be equivalent on both sides of the y axis as long as they are equal in terms of horizontal displacement from the y axis.

Some even functions

A function f is odd if, for each x in the domain of f ,

Odd functions are still symmetrical, but are symmetrical about the origin unlike how even functions are symmetrical about the y axis.

An odd function