What is the domain and range for the following function and its inverse? f(x)=(3x-1)/(2)?

Question

domain and range for the following function and its inverse

Answer 1

f(x) = ( 3x - 1) / ( 2 )

3x - 1  is not equal to 0 (because of f(x) will turn zero

3x - 1 = 0    Therefore  x  = 1/3

domain = x belongs to ( -infinity , infinity) - {1/3}

for range

f(x) = ( 3x - 1) / (2 )

Multiply each side by 2.

2 f (x ) = [ ( 3 x -1) / 2 ] 2

2 f ( x)  = 3 x - 1

Add  1  to  each  side

2 f ( x ) + 1  =  3 x  - 1 + 1

2 f ( x )  + 1 =  3x

Then  x  =  {2 f ( x ) + 1} / 3

{2f(x ) +  1}/3  is not equal to 0.

therefore f(x) is not equal to (-1/2).

range = ( -infinity , infinity) - {-1/2}

Let  f ( x ) = y

( 3 x + 1 ) / 2  =  y

2 y  =  3 x  + 1

(  2 y  - 1 ) / 3  =  x

 Inverse  function is f'(x) = (2x - 1) / 3.

Comments

Domain and range are all real numbers.

Answer 2

The equation is f(x ) = (3x - 1)/2.

y = 3/2 x - 1/2

Make the table of values to find ordered pairs that satisfy the equation.

Choose values for x and find the corresponding values for y.

x	y = 3/2 x - 1/2	(x, y)
-2	y = (3/2)(-2) - 1/2 =-7/2	(-2,-7/2)
-1	y = (3/2)(-1) - 1/2 = -2	(-1,-2)
0	y = (3/2)(0) - 1/2 = -1/2	(0,-1/2)
1	y = (3/2)(1) - 1/2 = 1	(1,1)
2	y = (3/2)(2) - 1/2 = 5/2	(2,5/2)

Draw a coordinate plane.

Plot the coordinate points.

Then sketch the graph, connecting the points with a line.

Since x can be any real number, there is an infinite number of ordered pairs that can be graphed. All of them lie on the line shown.

Notice that every real number is the x - coordinate of some point on the line.

Also, every real number is the y - coordinate of some point on the line.

So, the domain and range are both all real numbers, and the relation is continuous.