For each function below, determine whether or not it is one-to-one. If the function is one-to-one, find a formula for the inverse.

Question

a) f(x)=x³+1

 

b) g(x)=1-x²

Answer 1

For a function to be one to one it must

pass the horizontal line test.  For every

output value f(x) there is only one

input value x..  Or, for each value in

the range there is only one

corresponding value in the domain.

 

f(a) = f(b) only if a = b

 

a) f(x) = x³ + 1

Ths is a one to one function. 

 

Inverse:

1) replace f(x) with y

y = x³ + 1

2) Switch x and y

x = y³ + 1

3) Solve for y

y³ = x - 1

y = ³√(x-1)

4) Replace y with f^-1(x)

f^-1(x) = ³√(x-1)

 

b) g(x) = 1 - x²

This function is not one to one

because f(x) = f(-x)

 

Example:

g(2) = 1-4 = -3

g(-2) = 1-4 = -3

 

This will not pass the horizontal line test.