^{2}is not one-to-one. x= -2 and x=2 both give f(x)=4

^{2}where x>0 then f(x) is one-to-one.

^{2}where x>0.

^{-1}(x)

^{-1}(x)=-2/(2x-1)

^{x}then f

^{-1}(x) = log(x)

^{x}then f

^{-1}(x) = ln(x)

How do you determine if a function has an inverse function or not? Can you provide a detail example on how to find the inverse function of a given function?

Tutors, sign in to answer this question.

Ryan S. | Mathematics and StatisticsMathematics and Statistics

A function has an inverse if and only if it is a one-to-one function. That is, for every element of the range there is exactly one corresponding element in the domain. To use an example f(x), f(x) is one-to-one if and only if for every value of f(x) there is exactly one value of x that gives that value.

f(x) = x^{2} is not one-to-one. x= -2 and x=2 both give f(x)=4

We can make a function one-to-one by restricting it's domain. In the previous example if we say f(x)=x^{2} where x>0 then f(x) is one-to-one.

The function g(x) = square root (x) is the inverse of f(x)=x^{2} where x>0.

Algebraic functions involve only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power.

If an algebraic function is one-to-one, or is with a restricted domain, you can find the inverse using these steps.

Example: f(x) = (x-2)/(2x) This function is one-to-one.

Step 1: Let y = f(x)

y=(x-2)/(2x)

Step 2: solve for x in terms of y

y=(x-2)/(2x)

2xy=x-2 multiply both sides by 2x

2xy-x=-2 subtract x from both sides

x(2y-1)=-2 factor out x from left side

x=-2/(2y-1) divide both sides by (2y-1)

Step 3: switch the x and y

y=-2/(2x-1)

Step 4: Let y=f^{-1}(x)

f^{-1}(x)=-2/(2x-1)

Some non-algebraic functions have inverses that are defined.

sin and arcsine (the domain of sin is restricted)

other trig functions e.g. cosine, tangent, cotangent (again the domains must be restricted.)

If f(x) = 10^{x} then f^{-1}(x) = log(x)

If f(x) = e^{x} then f^{-1}(x) = ln(x)

Already have an account? Log in

By signing up, I agree to Wyzant’s terms of use and privacy policy.

Or

To present the tutors that are the best fit for you, we’ll need your ZIP code.

Your Facebook email address is associated with a Wyzant tutor account. Please use a different email address to create a new student account.

Good news! It looks like you already have an account registered with the email address **you provided**.

It looks like this is your first time here. Welcome!

To present the tutors that are the best fit for you, we’ll need your ZIP code.

Please try again, our system had a problem processing your request.

Fred B.

All Maths, Physics: Step through processes to experience / learn

$12.50 per 15 min

View Profile >

Suresh R.

All Math, Physical, Computer Sciences and Test Prep.

$12.50 per 15 min

View Profile >

Jeff R.

Calculus and Physics Specialist! 13 years experience

$11.25 per 15 min

View Profile >