Given the points (1, 12) and (-1, 0.75)
We will write the exponential function that includes the given points
Let the expression of the function will be as follows:
[tex]f(x)=a\cdot b^x[/tex]When x = 1, f = 12
so,
[tex]12=a\cdot b\rightarrow(1)[/tex]When x = -1, f = 0.75
So,
[tex]\begin{gathered} 0.75=a\cdot b^{-1} \\ 0.75=\frac{a}{b}\rightarrow(2) \end{gathered}[/tex]Solve the equations (1) and (2) to find (a) and (b)
Multiply the equations to eliminate (b)
[tex]\begin{gathered} 12\cdot0.75=a^2 \\ a^2=9 \\ a=\sqrt[]{9}=3 \end{gathered}[/tex]Substitute with (a) into equation (1) to find (b)
[tex]\begin{gathered} 12=3b \\ b=\frac{12}{3}=4 \end{gathered}[/tex]So, the answer will be:
The equation of the exponential function will be as follows:
[tex]f(x)=3\cdot4^x[/tex]