The function f(x)= 4x+2/9x-5 is one -to -one. Find its inverse and check your answer.

Question

f^-1(x)=

Answer 1

The rational function

• Vertical asymptotes are found  by solving denominator = 0

9x - 5 = 0

9x = 5

vertical asymptote x = 5/9

• To find horizontal asymptote.

Degree of the numarator = 1 and the degree of denominator = 1.

If the degree of the numerator is equal to the degree of the denominator, then horizontal asymptote y = ratio of leading coefficients.

y = 4/9 is horizontal asymptote.

Graph the function.

The function is one - to - one ,since there is a horizontal line that hits the graph at one point.

 

Answer 2

To find inverse

Change f (x) to y.

Interchange x  and y .

Solve for y .

Replace the variable y by the inverse notation f ^-1(x ).

Check

• Find the domain and range  of function

A rational function is simply fraction and in a fraction the denominator cannot be equal to 0 because it would be undefined.

To find which number make the fraction undefined create an equation where the denominator is not equal to zero.

Domain set of function

Range set of function

• Find the domain and range  of inverse function

Domain set of inverse function

Range set of inverse function

f  is one - one function with domain A  and range B  ,Then it's inverse function denoted by f ^-1 has Domain B and Range A  and is defined by f ^-1(y ) = x and ⇔ f (x ) = y  for any y ∈ B  .

Therefore the inverse is .