Using an exponential function, it is found that f(7.5) = 45.5.
An exponential function is modeled by:
[tex]y = ab^x[/tex].
In which:
For this problem, we have that f(3.5) = 12, then:
[tex]y = ab^x[/tex]
[tex]12 = ab^{3.5}[/tex]
[tex]a = \frac{12}{b^{3.5}}[/tex]
We also have that f(9) = 75, then:
[tex]y = ab^x[/tex]
[tex]75 = ab^9[/tex]
Since [tex]a = \frac{12}{b^{3.5}}[/tex]:
[tex]75 = \frac{12}{b^{3.5}} \times b^9[/tex]
[tex]b^{5.5} = \frac{75}{12}[/tex]
[tex]\sqrt[5.5]{b^{5.5}} = \sqrt[5.5]{\frac{75}{12}}[/tex]
[tex]b = \left(\frac{75}{12}\right)^{\frac{1}{5.5}[/tex]
b = 1.39542165
Then:
[tex]75 = ab^9[/tex]
[tex]a = \frac{75}{1.39542165^9}[/tex]
a = 3.7386
Hence:
[tex]y = 3.7386(1.39542165)^x[/tex]
When x = 7.5, we have that:
[tex]y = 3.7386(1.39542165)^{7.5} = 45.5[/tex]
Hence f(7.5) = 45.5.
More can be learned about exponential functions at https://brainly.com/question/25537936
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