Question

If f ( x ) f(x) is an exponential function where f ( − 3.5 ) = 25 f(−3.5)=25 and f ( 6 ) = 33 f(6)=33, then find the value of f ( 6.5 ) f(6.5), to the nearest hundredth.

Answer 1

Given:

f(x) is an exponential function.

`f(-3.5)=25, f(6)=33`

To find:

The value of f(6.5).

Solution:

Let the exponential function is

`f(x)=ab^x` ...(i)

Where, a is the initial value and b is the growth factor.

We have,
`f(-3.5)=25`. So, put x=-3.5 and f(x)=25 in (i).

`25=ab^(-3.5)` ...(ii)

We have,
`f(6)=33`. So, put x=6 and f(x)=33 in (i).

`33=ab^(6)` ...(iii)

On dividing (iii) by (ii), we get

`(33)/(25)=(ab^(6))/(ab^(-3.5))`

`1.32=b^(9.5)`

`(1.32)^{(1)/(9.5)}=b`

`1.0296556=b`

`b\approx 1.03`

Putting b=1.03 in (iii), we get

`33=a(1.03)^(6)`

`33=a(1.194)`

`(33)/(1.194)=a`

`a\approx 27.63`

Putting a=27.63 and b=1.03 in (i), we get

`f(x)=27.63(1.03)^x`

Therefore, the required exponential function is
`f(x)=27.63(1.03)^x`.