Question

If f(x) is an exponential function where f(3) = 18 and f(7.5) = 60, then find the

Answer 1

Given:

f(x) is an exponential function where f(3) = 18 and f(7.5) = 60.

To find:

The value of f(12), to the nearest hundredth.

Solution:

Let the exponential function be

$f(x)=ab^x$         ...(i)

It is given that $f(3) = 18$. Substitute $x=3$ in (i).

$f(3)=ab^3$

$18=ab^3$           ...(ii)

It is given that $f(7.5) = 60$. Substitute $x=7.5$ in (i).

$f(7.5)=ab^{7.5}$

$60=ab^{7.5}$           ...(iii)

Divide (iii) by (ii).

$\dfrac{60}{18}=\dfrac{ab^{7.5}}{ab^3}$

$\dfrac{10}{3}=b^{4.5}$

$\left(\dfrac{10}{3}\right)^\frac{1}{4.5}=b$

$b\approx 1.30676$

Putting $b=1.30676$ in (ii), we get

$18=a(1.30676)^3$

$\dfrac{18}{(1.30676)^3}=a$

$a\approx 8.0665$

Putting $a=8.0665$ and $b=1.30676$ in (i), we get

$f(x)=8.0665(1.30676)^x$

Putting x=12, we get

$f(12)=8.0665(1.30676)^{12}$

$f(12)=200.0024$

$f(12)\approx 200.00$

Therefore, the value of f(12) is about 200.00.