FUNCTIONS and Domains...10pts?

Question

The given function f is one-to-one. f(x)= (ax+b)/(cx+d) Find the domain of f

The given function f is one-to-one. f(x)= (ax+b)/(cx+d) Find f

The given function f is one-to-one. f(x)= (ax+b)/(cx+d) Find the range of f

Answer 1

Dear Sharlette;

Part 1 ;
f(x)= (ax+b)/(cx+d)
In the fractions function . the denominator shouldn't be equal zero .b coz it makes the fraction undefined.
cx+d = 0
cx = -d
x = -(d/c)
Domain = R - { - (d/c) }
Or
all set of real numbers EXCEPT " -d/c "
Or
( -∞ , - d/c) U ( -d/c , +∞ )


Part 2
let y = (ax+b)/(cx+d)
y * ( cx + d) = ax + b
ycx + yd = ax + b
ycx - ax = b - yd
{ Factor ' x' }
x ( yc - a ) = b - yd
x = ( b - yd) / ( yc - a)
Now swap x & y
y = ( b - xd ) / ( xc - a)
It is inverse function.

Part 3
Range ;
for fiding range you should find the inverse function , so we did in part 2
domain of inevrse function is the range of main function ;
y or f^(-1) x = ( b - xd ) / ( xc - a)
xc - a = 0
xc = a
x = a/c
Range = R - { a/c}
OR
( -∞ , a/c ) U ( a/c , + ∞)

Good Luck Darling.

Answer 2

The domain of (ax+b)/(cx+d) consists of all x such that cx+d is nonzero. Solve for x to describe the domain precisely.

For the second one, I assume you mean the inverse function f^-1, not f. To find it, set y=(ax+b)/(cx+d). Solve for x by multiplying both sides by (cx+d) and then isolating all terms with x on one side of the equation, with all other terms on the other side. Then interchange x and y to obtain f^-1(x).

For the last one, use the fact that the range of f is the domain of f^-1. Find the domain of f^-1 using your answer to the previous part, and the approach of the first question.

Answer 3

the domain is:
cx+d not cero
so, if c is not cero that means that
x has to be different from -d/c
but if c=0 it means that d would be different from cero and the domain would be all the real numbers.

the range of the function is all real numbers

and the inverse of the function is:

f^{-1}(y)=(b-yd)/(cy-a)

Answer 4

domain of f: (-ﻩﻩ, -d/c)U(-d/c, ﻩﻩ)
range of f: (-ﻩﻩ, a/c)U(a/c, ﻩﻩ)

f^(-1)=(b-xd)/(xc-a)