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

$\dfrac{33}{25}=\dfrac{ab^{6}}{ab^{-3.5}}$

$1.32=b^{9.5}$

$(1.32)^{\frac{1}{9.5}}=b$

$1.0296556=b$

$b\approx 1.03$

Putting b=1.03 in (iii), we get

$33=a(1.03)^{6}$

$33=a(1.194)$

$\dfrac{33}{1.194}=a$

$a\approx 27.63$

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 $f(x)=27.63(1.03)^x$.