Inverse Functions

[nycmathguy]

A function f has an inverse
function if and only if f is one-to-one.

Based on the above statements, do the following 5 functions have an inverse?

A. f(x) = x^2

B. f(x) = x^3

C. f(x) = | x |

D. f(x) = sqrt{x}

E. f(x) = 1/x

 

[MathLover1]

A function f has an inverse function if and only if f is one-to-one.

A. f(x) = x^2

to find inverse, first note that f(x) =y
y) = x^2 ...............swap variables
x= y^2
y=

=>one-to-one

B. f(x) = x^3
y= x^3
x= y^3
y=

=>one-to-one

C. f(x) = | x |

There is no inverse => not one-to-one

D.
f(x) = sqrt(x ) inverse is
y= sqrt(x )
x= sqrt(y )
y=x^2=> not one-to-one

E. f(x) = 1/x
inverse =>
y = 1/x...............swap variables
x = 1/y...............solve for y
y= 1/x=> is one-to-one

 

[nycmathguy]

A function f has an inverse function if and only if f is one-to-one.

A. f(x) = x^2

to find inverse, first note that f(x) =y
y) = x^2 ...............swap variables
x= y^2
y=

=>one-to-one
View attachment 252

B. f(x) = x^3
y= x^3
x= y^3
y=

=>one-to-one
View attachment 251

C. f(x) = | x |

There is no inverse => not one-to-one

D.
f(x) = sqrt(x ) inverse is
y= sqrt(x )
x= sqrt(y )
y=x^2=> not one-to-one

E. f(x) = 1/x
inverse =>
y = 1/x...............swap variables
x = 1/y...............solve for y
y= 1/x=> is one-to-one

Another easy reply.