Question

Which statements accurately describe the function f(x) = 3(16) Superscript three-fourths x? Select three options.

The initial value is 3.
The domain is x > 0.
The range is y >0.
The simplified base is 12.
The simplified base is 8.

Answer 1

The true statements include

- The initial value is 3.

- The range is y >0.

- The simplified base is 8.

Explanation:

f(x) = 3 (16^0.75x) = 3 (16⁰•⁷⁵ˣ)

We will check each of the options for how true they are.

- The initial value is 3.

The initial value of a function is when x=0

and when x=0

f(x) = 3 (16⁰•⁷⁵ˣ)

At x = 0

f(x) = 3 (16⁰) = 3 × 1 = 3.

This statement is true as the the initial value of the function is indeed 3.

- The domain is x>0

The domain of a function refers to the region of values of x where the funcfion exists.

f(x) = 3 (16⁰•⁷⁵ˣ)

It is evident that f(x) will exist any where for any real number value of x. Especially for regions where x ≤ 0, contrary to this statement. Hence, this statement is false.

- The range is y >0.

The range of a function is the set of numbers or region or interval of values that the function can take on.

f(x) = 3 (16⁰•⁷⁵ˣ)

It is evident that this function will always be positive. Hence, the range is truly y>0.

This statement is true.

The last two statements will be solved similarly

f(x) = 3 (16⁰•⁷⁵ˣ)

To simplify This,

f(x) = 3 (2⁴)⁰•⁷⁵ˣ = 3 (2³ˣ) = 3 (8ˣ)

The simplified base is evidently 8.

- The simplified base is 12.

This statement is not evidently true.

- The simplified base is 8.

This statement is true.

Hope this Helps!!!

Answer 2

The initial value is 3.

The range is y greater than 0.

The simplified base is 8.

Step-by-step explanation:

The given function is
`f(x) = 3(16)^{(3)/(4)x }` ......... (1)

Therefore, the initial value of the function at x = 0 is
`f(0) = 3(16)^(0) = 3`

Now, the domain can be any real value, since for all real value of x, y exists.

But, for no value of x the function has value < 0.

Therefore, y greater than 0 is the range of the function f(x).

Now, simplifying the equation (1) we will have

`f(x) = 3(16)^{(3)/(4)x } = 3(16^{(3)/(4) } )^(x) = 3(8)^(x)`

Therefore, the simplified base is 8. (Answer)