Question

inverse function property to show that f and g are inverse

f(x)= (x-5)/(3x+4) and g(x)= (5+4x)/(1-3x)


please help!

Answer 1

I love these questions

Answer 2

To find the inverse of a function change x to y and change y to x and then solve for y

for example the function:

f(x) = 3x - 1

to find its inverse we switch variables

y = 3x - 1
switch variables:
x = 3y - 1
solve for y

x + 1 = 3y
(x+1)/3 = y
therefore the inverse is:
\[f^{-1}(x) = \frac{(x+1)}{3}\]

Answer 3

O shizzzz I think he has the floor

Answer 4

Note: Not all functions have an inverse, for a function to have an inverse it must be one-to-one. To check for this property there is a thing called the horizontal line test see:

http://www.mathwords.com/h/horizontal_line_test.htm

Essentially for a function to be one to one, every input (x value put into the function) must have a unique output (y value provided as a result of the inputted x value out by the function)

for example:
A function containing the points:

(3, 1) and (4, 1) would not be one to one because there are two x values that give the same output

Answer 5

sorry for not letting you answer this one Kagome9

Answer 6

Didnt realize you were going to answer it :)

Answer 7

\[(f\circ g)(x)\\
=f(g(x))\\
=f\left(\frac{5+4x}{1-3x}\right)\\
=\frac{\left(\frac{5+4x}{1-3x}\right)-5}{3\left(\frac{5+4x}{1-3x}\right)+4}\\
=\frac{\left(\frac{5+4x}{1-3x}\right)-5}{3\left(\frac{5+4x}{1-3x}\right)+4}\times\frac{1-3x}{1-3x}\\
=\frac{\left(5+4x\right)-5(1-3x)}{3\left({5+4x}\right)+4(1-3x)}\\
=\frac{5+4x-5+15x}{15+12x+4-12x}\\
=\frac{19x}{19}\\
=x\]


\[(f\circ g)(x)=x\iff g= f^{-1}\]