# If f(a) is an exponential function where f(-3) = 18 and f(1) = 59, then find the value of f(0), to the nearest hundredth.

If f(a) is an exponential function where f(-3) = 18 and f(1) = 59, then find the
value of f(0), to the nearest hundredth.

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1. Given:

For en exponential function f(a):

$$f(-3)=18$$

$$f(1)=59$$

To find:

The value of f(0).

Solution:

The general form of an exponential function is:

$$f(x)=ab^x$$          …(i)

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

We have, $$f(-3)=18$$. Substitute $$x=-3,f(x)=18$$ in (i).

$$18=ab^{-3}$$            …(ii)

We have, $$f(1)=59$$. Substitute $$x=1,f(x)=59$$ in (i).

$$59=ab^{1}$$            …(iii)

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

$$\dfrac{59}{18}=\dfrac{ab^{1}}{ab^{-3}}$$

$$3.278=b^{1-(-3)}$$

$$3.278=b^{4}$$

$$(3.278)^{\frac{1}{4}}=b$$

$$1.346=b$$

Substituting the value of b in (iii).

$$59=a(1.346)^1$$

$$\dfrac{59}{1.346}=a$$

$$43.83358=a$$

$$a\approx 43.83$$

The initial value of the function is 43.83. It means, $$f(0)=43.83$$.

Therefore, the value of f(0) is 43.83.