Question

If f(x)f(x) is an exponential function where f(3.5)=12f(3.5)=12 and f(9)=75f(9)=75, then find the value of f(7.5)f(7.5), to the nearest hundredth.

Answer 1

Using an exponential function, it is found that f(7.5) = 45.5.

What is an exponential function?

An exponential function is modeled by:

$y = ab^x$.

In which:

• a is the initial value.

• b is the rate of change.

For this problem, we have that f(3.5) = 12, then:

$y = ab^x$

$12 = ab^{3.5}$

$a = \frac{12}{b^{3.5}}$

We also have that f(9) = 75, then:

$y = ab^x$

$75 = ab^9$

Since $a = \frac{12}{b^{3.5}}$:

$75 = \frac{12}{b^{3.5}} \times b^9$

$b^{5.5} = \frac{75}{12}$

$\sqrt[5.5]{b^{5.5}} = \sqrt[5.5]{\frac{75}{12}}$

$b = \left(\frac{75}{12}\right)^{\frac{1}{5.5}$

b = 1.39542165

Then:

$75 = ab^9$

$a = \frac{75}{1.39542165^9}$

a = 3.7386

Hence:

$y = 3.7386(1.39542165)^x$

When x = 7.5, we have that:

$y = 3.7386(1.39542165)^{7.5} = 45.5$

Hence f(7.5) = 45.5.

More can be learned about exponential functions at brainly.com/question/25537936

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