Answer:
[tex]f(x) = 7\, ((1/2)^{x})[/tex].
Step-by-step explanation:
An exponential function is typically in the form [tex]f(x) = a\, (b^{x})[/tex], where [tex]a[/tex] and [tex]b[/tex] ([tex]b > 0[/tex]) are constants to be found.
In this question:
[tex]f(2) = 1.75[/tex] means that [tex]a\, (b^{2}) = 1.75[/tex].
[tex]f(-2) = 28[/tex] means that [tex]a\, (b^{-2}) = 28[/tex].
Divide one of the two equations by the other to eliminate [tex]a[/tex] and solve for [tex]b[/tex].
The number of the right-hand side of the second equation is larger than that of the first equation. Hence, divide the second equation with the first:
[tex]\displaystyle \frac{a\, (b^{-2})}{a\, (b^{2})} = \frac{28}{1.75}[/tex].
[tex]\displaystyle b^{-4} = 16[/tex].
[tex]b^{-1} = 2[/tex].
[tex]\displaystyle b = \frac{1}{2}[/tex].
Substitute [tex]b = (1/2)[/tex] back into either equation (for example, the first equation) and solve for [tex]a[/tex]:
[tex]a\, ((1/2)^{2}) = 1.75[/tex].
[tex]a = 7[/tex].
Substitute [tex]a = 7[/tex] and [tex]b = (1/2)[/tex] into the other equation. That equation should also be satisfied.
Therefore, this function would be:
[tex]f = 7\, ((1 / 2)^{x})[/tex].
