Question

Which function has a range of y < 3?

A. y=3(2)^x
B. y=2(3)^x
C. y=-(2)^x+3
D. y=(2^x)-3

Answer 1

The correct option is C.

Explanation:

If a function is defined as

`f(x)=a^x`

where, a>0, then the range of the function is greater than 0.

`a^x>0` .... (1)

Option A: Using inequity (1),

`(2)^x>0`

Multiply both side by 3.

`3(2)^x>0`

`y>0`

The range of first function is y>0. Therefore option A is incorrect.

Option B: Using inequity (1),

`(3)^x>0`

Multiply both side by 2.

`2(3)^x>0`

`y>0`

The range of second function is y>0. Therefore option B is incorrect.

Option C: Using inequity (1),

`(2)^x>0`

Multiply both side by -1.

`-(2)^x<0`
`(\text{change the sign of inequality})`

Add 3 on both the sides.

`-(2)^x+3<3`

`y<3`

The range of first function is y<3. Therefore option C is correct.

Option D: Using inequity (1),

`(2)^x>0`

Subtract 3 from both the sides.

`2^x-3>-3`

`y>-3`

The range of second function is y>-3. Therefore option D is incorrect.

Answer 2

Using a graphing tool

Let's graph each of the cases to determine the solution of the problem

case A)
`y=3(2^(x))`

see the attached figure N
`1`

The range is the interval--------> (0,∞)

`y> 0`

therefore

the function
`y=3(2^(x))` is not the solution

case B)
`y=2(3^(x))`

see the attached figure N
`2`

The range is the interval--------> (0,∞)

`y> 0`

therefore

the function
`y=2(3^(x))` is not the solution

case C)
`y=-2^(x)+3`

see the attached figure N
`3`

The range is the interval--------> (-∞,3)

`y< 3`

therefore

the function
`y=-2^(x)+3` is the solution

case D)
`y=2^(x)-3`

see the attached figure N
`4`

The range is the interval--------> (-3,∞)

`y>-3`

therefore

the function
`y=2^(x)-3` is not the solution

the answer is

`y=-2^(x)+3`