Find the inverse function f^-1 and state its domain. f(x)=(x^2-1)/(x^2+1), x ⩽0

Name: Problem 23, Chapter 2: Higher Level Mathematics
Uploaded: 2022-01-31T15:50:55-08:00
Duration: 3 min 12 s
Channel: Gaurav Kalra
Description: Find the inverse function f^-1 and state its domain. f(x)=(x^2-1)/(x^2+1), x ⩽0

step 1 I am the given caution we're given FX with a given domain. We need to find out the inverse funct

Find the inverse function f^-1 and state its domain....

Key Concepts

Inverse Function

An inverse function essentially reverses the action of the original function. If f maps an element x to an element y, then its inverse f?¹ maps y back to x. In mathematical terms, f(f?¹(y)) = y and f?¹(f(x)) = x. Finding the inverse typically involves swapping the roles of x and y and solving for the new output.

Domain and Range

The domain of a function consists of all valid inputs, and its range is the set of all outputs. When a function is inverted, the original function’s range becomes the domain of the inverse function, and the original function’s domain becomes the range of the inverse. Understanding this relationship is key to correctly determining the domain for an inverse function.

Restricted Domain for One-to-One Functions

A function must be one-to-one (injective) in order to have an inverse that is also a function. In many cases, a function is restricted to a subdomain to ensure it is one-to-one. This restriction is crucial because without it, the function might map more than one input to the same output, making the inverse ambiguous.

Solving Equations to Find the Inverse

Finding an inverse function involves algebraically manipulating the equation to interchange the dependent and independent variables. This process often requires careful attention to operations like taking square roots, where additional conditions (such as sign restrictions coming from the original domain) must be applied to ensure the inverse function is correctly defined.

Transcript

00:03 I am the given caution we're given fx with a given domain.

00:08 We need to find out the inverse function and then we need to state its domain.

00:16 So if we just go with that, we're given fx is equal to x square minus 1.

00:23 So fx equals x square minus 1, all divided by x square plus 1, x2 plus 1, right? in the domain, for this function is given as x is less than equal to 0.

00:42 Now to find the inverse function, what we'll do, we'll put y equals x square minus 1 over x squared plus 1.

00:50 Next step we'll do, we just replace x and y, so we get x equal to y square minus 1 over y square plus 1, and then find the value y from here, so we get x y square plus x equals y square minus 1.

01:05 So we saw this, we get x minus 1, y square equals, we get from the, we get from here minus x minus 1 or we get y squared equals minus x minus 1 over x minus 1 or we get x plus 1 over 1 minus x right multiplying the numerator denominator with minus 1 so uh now we have here for the negative domain x is less than equal to 0 right so this is domain here and that domain become the range so y value is coming out to be we did the negative value now here not positive values it will be negative root x plus 1 over 1 minus x this is we get the inverse function so we get f inverse x is coming out to be so f inverse x is equal to we will have minus square root x plus 1 over 1 minus x that's coming hard to be the inverse function now to find domain of this so domain is we have this one square root that means this part here and that a square root that should be greater than equal to 0.

02:17 So we have x plus 1 over 1 minus x that should be greater than equal to 0.

02:23 So from here you find out domain.

02:26 So we draw here sign diagram.

02:31 So we have x equal to 1...

Please give Ace some feedback