let f:R from R be a function given by f(x)=x^2+1 find f inverse {-5}, f inverse{26}?

Answer 1

First find the inverse:

x = [fˉ¹(x)]² + 1
x - 1 = [fˉ¹(x)]²
fˉ¹(x) = √(x - 1)

(Could also have fˉ¹(x) = −√(x - 1)...but not both... or it won't be a function...)

fˉ¹(5) is undefined (can't take the square root of a negative number). This makes sense, because it's the equivalent of trying to solve -5 = x² + 1, which has no solution.

fˉ¹(26) = √(26 - 1) = √25 = 5 (or -5, depending which inverse function you choose).

Answer 2

f inverse -5 does not exist as the function is >0 for all x and f inverse 26 root of 26-1=5 the inverse function is root of (x-1)

Answer 3

It facilitates to write down y=x^5+5x+a million To get the inverse function you purely substitute the x and y, so x=y^5+5y+a million and once you attain expressing y in terms of x, you have the inverse function.

Answer 4

f{x} >0
because
X^2>0
1>0
x^2+1>0
f{x}>0
there fore

-5 does not lies in you function

and the aother
one
is
26=X^2+1
26-1=x^2
25=X^2
x=+5 or -5

ther fore f inverse of {-5} does not exists,
f inverse of {26} is -5 or+5

Answer 5

let,
x^2+1=y
x=(y-1)^1/2
therefore
f inverse(-5) is imaginary=6i
f inverse(26) is=5