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